GROUPS OF NEAT AND PURE-HIGH EXTENSIONS OF SOME ABELIAN GROUPS THESIS SUBMITTED EOR THE DEGREE OE Doctor of Philosophy IN MATHEMATICS By ASIF MASHHOOD DEPARTMENT OF MATHEMATICS FACULTY OF SCIENCE ALIGARH MUSLIM UNIVERSITY, ALIGARH 1982
GROUPS OF NEAT A N D PURE-HIGH EXTENSIONS OF
SOME ABELIAN GROUPS
THESIS SUBMITTED EOR THE DEGREE OE
Doctor of Philosophy IN
MATHEMATICS
By
ASIF MASHHOOD
DEPARTMENT OF MATHEMATICS
FACULTY OF SCIENCE
ALIGARH MUSLIM UNIVERSITY, ALIGARH
1 9 8 2
mammmmmmm
1 M r * t1ii^ •ppiTtiBiitjr • f miptfmwtMg W9 ^—P coi^)*
• f «ratiti€« to l a t t Iff f•mm M*A» I w i s , tli« ()Bm«p BasA,
BtpwnuBt af i to^«* t io9 , AligtHi VMl ia Vmirmaitf^ « I M
inaii lU af Id * i i i i f f « « i t iMalHi iBtroteetA M f t h i *
Ivaacli af Alfttoa aaA •»&• « • ae^atntad wltli tha ttt« 4«r«l«|»*
•« i i « la tiM t^Mipy of atal i ta grovpa* B* ««« a «aiv«a af
graat iaa^at io t t ta »a a l l tht %iaa tbat X was aiigi«aA in
%hi9
X aa aa laaa l»diftta< ta Brafaaaav 9rat Xsltar flasaliit
BaaAfBapartBant af llatluttatiaatAlidarli INialte ttelrarsitar fue
praridlBg »a a l l Hkt aaaaaaary fa^aiUtiaa far tlila aartc* Sia
Hraaaal la1i«>aat Ja aer v«rtc \if wt^ af aaaaaragaawit IMMI baaa
• f laMBaa ralaa ta aa*
Vr tlMHiea ara alaa daa ta ay aallaagaaa la tlia 4apart*
Mat t lAa mmm^gH aa la paraalac tlda yra^aet.
I«atly X aa al^a graatftil ta ttr. latieia All IlMa* «AM
laapita af Ida \mtgt ««rtt aakatelat oaaU aiiara tha tlaa far
tjrplac rmj aaapataatlsr ay tkaaia*
S l ^ ' S a a U a M v a r a i ^ ^ ^^" lASHHOOD ) numh • 20t001«(lBiia ) lteta« Jaaa* 1982.
tiXM Of OORTXITS *iMMMMMMMMMMMMMMk*
l!MtflM* ••• ••• ••• ( i )
t* lBi«l ••wiseti i mA •xtansieas ••• 9
9« BigH «xt«iuilM< •• • 17
1. ffeui grettp • • ' tp ••• 29
2» ni* gvMip B*xl ••• 91
9* !••% «i4 fir««ldili «x%ffifliUiit • f » tersies* uNW ipravp ••• 99
4* I««t MA 9«r««lii|^ txt«R«i«ii« ttf a ttrsiea •^••l^ ••• 44
1. l i t««t tt««flieM ••• 92
2* 9«1vro«99 and <ni«fti«it «r««v« ••• C9
9» e«vpfKt«tl«i «f Viait(B*A) ••• f8
C1Wi»%tr»IVt Jfaolp WA fit "* JBUOttft*
1* SOM IfHHM ••• 7 i
(11)
2« Btzt * groups ••• 81
5* ^ - groups ••• 33
Chapter-^ t ? irf*Mfi •' ^ n«*t-Mgli axtgnsloas
1« Vsat-blgh txtanslons ••• 96
2* 1f*at a&d purs sxtanslons ••• 102
• • • • • • • • • • \ 1 /"^ XT /
?R ^ r # f f y
RsHolegleal alftttra in rM«nt d«ead«« hes fsrmm
«9 ta tM • vofit isp«rt«Bt tool In th« study of dif fortnt
fftneturos* Tt « M dlMOOYtPod for t ^ f irst tin* Yy R«rri«on
i ( I that bowoloiEioal oli^olro pla^a «n i«p«*t{int rolo in ttoo
tlioery of a^olion «roiip9« Mo uw& honologiool eothodo in tlw
diooorary of eroopa of oxtoBniennt puro oxtonsions, ootoroion
(crovpo oad odjuattd ^oupo* Horriooa | 8 ) Offoln oxploitod
hOROlOffiool olgotra to inT»«tigoto tlio proportion of tho
bifflit piiro<4iigh and noat^ifb oxtwaiona. lAtar othMr
•atlioMitioiaaa aoeh a« Paoha i 4 i » Zrola )l9it ^Ikar 127U
Talqrft |89|f llaaffaoaaaor |22{, BanaVdnlla I 1 i and mmaj otliara
adopt ad ttoaologleal toottniqaon in tho daraloiaiMt cf tho
atalian group thoory.
Tlio atady of tha ^etip of oxtonaiono» para axtaaalona*
naat axtanaiont*, p«ro-4iiirli «id Boat«id«li axtoailona of a
lereiip A ty a greap B dapwdo vainly on tho froapa A and ?•
NimaFOUii propvtiaa of th^a axtanaiono, in tha aVoTO aaatiottad
(11)
pmpmw «ir« •8t«Vliiih»4 \j ««la^ honologieal aathoda amd.
If obnfiBi th« fcrovpff A find B. <^pllttlng axtvialoiit
pla^ an iBpcrtmt rol* la tb« ntudy Pt dlffartnt tgrp A
•f 0xt«islons,
Tli« Miln teplm af dlncnRflion in thia tli««i« art tha
aaat and pvra-toiich axtwAlena and tha grenpa llazt(B»A) and
Bast (B,A) for a«Ka aballan grenpa A and B, JUtimtmt
tjpaa af abaliaa groapa ara anad and bcvolegleal vatboda
md taahnioaas ara avpleyad far tha davalepnant af tlia
raanlta* Tliroagli aat thia thaala al l groapa ahieli hav»
\9m ooRffidirad T* atalian*
fhla thaeis la dlvldad iata flTa ehaptara. Tba pris*
elpal pwrpaaa of tba introdaetflrj eliaptir X on prallal*
aaria9, la to aa^alat t)ia raadtr altli tha tarvlaolagjr aad
tha taala raaalta of atalian groap thaarj^ whloh ara ofton
uaad in tha av(b«a(|Baat ahaptara. Thia ahaptar la alaa
IntMidad ta aiira tha thaala en meh nalf-aaatalnad aa
poanltla* Hira wa hava givaa aaaa daflnltloaa and p r o p v
tlaat spealally af dlTloltla groapot para, naat oii^sroapa
( i U )
pw^0i end AMt-liigh txtrntiiefis ere «l«o dtfisad.
and R«xt^(ltA) \j vmgrtnt th* ^oups A end f frm tordieii
to t«r«ioB«ft»«« ftroQii** It vilX t« t^mttfA that i f B is
• t«r«io« group th«a th« t uMrioar of tho group Host (B«A)
io olvoift tho mmtf mm %tmt of lioxt(C,A) for ony group C.
l«t i f A i« ft tor«ioi»-froo group Msd B i9 o torfilini groupt
thai i t will l$o found thot tlio tvo irroupo Roi:t(B«A) and
Hoxt (BftA) do not t«liitTO in ft slviiar uay* For infitmeot
Soxt( C/ZyA} iii « roduood ftlgotrftioftlljr eoepeet group vhilo
Roxt (C/S«A) • 0, flion fro lioTo fftudiod tlio groups ltoxt(BtA) p
ftod Rost CB,A) kooping A ft toPAion«»ft*oo group oad iiipooiag p
no rootriotion on tht roup B* It will to oVAWTOd that tho
two groupo fti^in do not Vohi vo in tho oaiso fashion* Pinalljt
«o h«TO o«tani«h«d that i f Veth tho /groups A «nd B ftro
toritlon groupuy tho groupo W*xt(B,A) and B«Kt (B,A} IMITO
oiiiilar iMTOpcrtioo*
Xn dinptor IXI« tho |i;roup Woxt(BtA) and Hast (F,A)
ha« YBm wtudiod Vy Torylng tho leronpo A mA f fro« oyelio
(IT)
grmip9 9t isrlflit ora»r to dlrtet nvm of ejrelie group* of
ivlao ontf priwo pmtr ordop, ibt»et • •(joencoff oormtetinir
^KKt to novCthooTM ?*2) and r«letliig Ho«* to Bxt to ??ort
(thoeriK 3*4) baro 1>o«ii o'ttotlishtd* 1%o «iit)grottp9 of
WcxtCByA), i s omo A io a tfirt«t SOP of ojrolio laroupo of
priwo poo«r ordtr IIOTO t)««n oeaotntotad md tbo proportioo
of tli»i«o 9«it>grovp« and tholr Qvotimt groiip<9 hetva 'bani
f«t«iAia4«
SB ehaptfr IT» wo Havo oone«ntr»to4 on finding out a
itroop 0 that la laovorphlo to tho irenp of pttra«4)ij?)s txtan*
•ilonit for aultaVl* je oopa A and B. ^ob grtnvn hoTa V^w
nanad as Baxt^fiarovpa. It haa Yoaa <ihoPB tliat an alfBiatarjr P
p-isroup i«a Haxt •uronp. Alao, al l th»»a irrowpa ittoona pora-
Mi^ axtwialena \y torsion flrroap^ M*a apllttlng (tluit In
H]: • irottpa) )iaT9 taan ^tndlad. <loffO of tba Important
, c « p « « « . . f • ; - ,re»w ! » « . U . t . . rM.r««I i , th l .
oliaptar*
Chaptar ? la davotad to atudsriAg tha pnraHsiih and
aoat'-hlgli axtan«iO!i«* Ooi^ltloaa vadv tfvhleb a f •«aat» hlerti
( T )
•xtanalon radnoM to a f«fwrt*hi(^ •xtcnalon fmd m nmt
•xttnuion oelneld«« with m pur» »xten«ioii hPTt "bMtn dl9»
oti9ift<l» <?pllttiBff oendltloan for naat and ptvfli ifh axttn*
Aiena, oendltion on i( and B tnidar vleh n«xt(B»ii) • 0 and
Htxt (B, Ji) M 0 havt ^aan dadueod and dl9etts«>ed.
PRSIIMISARISS
Th* B«ln fNtrpoftft of this Introdvotory ehApttr i s to
r tes l l 9om9 ef tha d^flnitlotiiit F«mlt« aad other nooviattiry
"Mta of Infemotlen ivhidi wil l l o ««i«4 fir^ouontly in tho «ttt*
i90i|«Mit ohoptors* fhitt i9 Voiniar dono enljr to fix up tbo ttrf«i*
nologf and nototienfi Btod»d in tho noqvtl and to mak9 thin «^k
<i«lfeont«in«d o« far oo po^iiiVla* Tho eontonts of this ehaptw
•ro net original and we«t of tha dafiniti<nia (SRd raffult* ara
tttm i 4 It! i t and I 8 U
In «iactioii 1,aoffa I'aflfie ooneapta dafinitiotKitf>l«rantary
propfrtiaa and raaaltat vainly of divi«tit:la ttvltaupwxjm, imra
autf^oupa and naat aiit groapa ara ii^ao*
In aaation 2,varieiia typm ef aanet na^aaeaa* eoetrutatiYa
diai^aamt naoa«ift«ry t i ta of infiMnratioB « d raeiilt«i for tha
^mipa of axtanftion;^ end pure axt«)fliionat tlareni^ hoaoloiB ioal Q
cathia ara ffivan, Tha important ooneapta of elgttraleally
eoKpaet and ootornien groop^ aith thair al«B«tary iropartiaa
any al<*e la fe«nd in thin taoticn.
In 9«etlo]s 3» hli?h,por«-hlgh, ntat and ntat-hif^h
•xtanAionn era dtfln«d. ' ovt of th« known ta<9ie r«iittXtfl
vbie)) Hill I t «4(>d In th« «uVn«cti«nt ebeptcr^ ar« stated
vltbout proof*
1- ' evo llM>ontn]*3r CoBoapts.
DoflnltionC 1.1 ).Qroupii la wM<* ovapy olortBt hat e
finito ordar art oallad tomlon ^otspa* thofft in whidh a l l
tho olwanta ^ 0 ara of Inflrita erdara art oallad top<«lon*
tt99 fifroupff*
I)afinitiQn(1.2)>A i;roup In whl«h tha ordarn of tha
alavan^ ara po»ar» of tha naaia prirea nuirtiar p la oallad a
privary or p<-f70iip*
Propo8itloa(1.9}« tvary torsion grovp i« a dlraet 9vm of
Pl^oops for variotta pi P,
PropoaItion( 1.4)>If F^ la tha naxisal tOTAlon mil group
(teraion part) of B, than S/B^ la toralon-fraa*
Pafinitionf 1 >5)>0roupa In vhldi arary alaaant haa Aouara-
flraa ordar ara oallad alanantary.
PropQwltifinf 1 .<).»vary ftlaKantary p-^oup i« a direct
mm of ogrollo Kronpa of tha tiima ord«r p»
Propoaitioaf 1.7). i i^oup > i» alavantary i f and only i f
arery «iuV«3TO«p of A isa diraot sunaiand.
Pafinltion(1*'^).Tha i>!rattlni mtt^fiymv cpiO) ot G ia tha
Intarnaetlon of all th» vaxlval "uVarouw* of 0, that la .
»
silsk faotior oX ^ an& I s afiKEio&u4 b>* ^ •
ijBfliiftaJJiifia(l*12). A lo olowataapy If and only If A l»
tile ooljr o&adaUal auO^rdui^ ia A«
iiSJUMLiAaA(^^4* Of a fa?©« aOallaa jraap l a aaont a
dlrt^ot aud of In f in i te oyoUo jr^upa* I f tnaaa o^ollo ^roupa
a re udner.v$ad by eloiaenta x^ilcl)« tban the fj?ea c<^t^ ^ wil l
be«
l a l^ *^
iaUOkliaBi^M)* A os:^up (# l a otOled fiULvlalOxa If
mi m a for a l l asii.
I t foUova fron ttie aafinl^lon «hat an eplflioc-.iilo laai^e
of a dlTlalble ;> aup l a 41irlalble and a dlreot iNa or a dlreot
^ coduot of «j;roup8 l a a lv la lb l^ 11 ind only If aaeH ooat^omms
l a alvlalola.ji^i£the£«oro. If «^^ilcl) are d lv is io ie &ub^£OMiia
of A, %htm m9 in th t i r mnm tUm*
petition 0 • 40 ^ R , wbaro 40 i s tha nexisel divinitlo 9ti1'«
group of Q* If 48 « 0, thon tho ureup 'S in eellod roduoo4.
pgepo#itio^1«t6).A 4iTifiiVlt «Btpro«p D of « frronp A
i s • diroet evpvmnA of A* tbat iPi ^ • I) 0 C for «o«o «iuV»
fsroup C of A*
Propooitionf 1«17).gyory group ooa to i Voddod •« • jwil-
Ijproup in o diYiffiVlo foroup*
Dofinition(1.18).A viniftol 4iTiiiillo group B oentoising
a gro«p A ifl e»llo4 tho divloit lo h«ll(iaj^etivo hull) of A.
^fiaition(1,f<l).A suVgrottp H of a i« oolltd a piiro
ou^grovp i f tlia oquotieii nx « 1i,«h«o im^tOgS i s dolvetlo
in H vhonoTir i t i t oolYot:ilo in G,or oonivoloiitlyt i f ,
nR a R O BO
itol4ii for oi l sotwol n«ct«p« n*
I t in ooajr to varify thftt orory diroet mRBJotid io o
pvro «iitgroup* 0 aad A art POPO mit^ottpfi of A« Tho tomion
port of • ttixod tpponp and it» p-oof>poiiaT)t<i aro uuro ant*
grovpa. Thaao fail in gtmxsciA, to %a dli?a«t awenandn. It io
to %9 notod that Q « 4 HxT) Imra no tmro fluV^oup«« la
twaioQ-firac ^oupa« iit«p«t«tioii of pw?a »»«Ttp»©U7 la again
pttro. Purity ia an induetira propartj.
Th«fr«E(1.20)^L»t T9C Vt niiVi ovpfl of A »roch thet
C ^ B £ A» th«n «t b«T««
(1) i f C iff p«r« in BfAnd F in pur* in A* th«i C id
piirt In A,
(2) i f B i s pur* in /» tb«n B/t! i s por« in VC»
(5) i f C i9 pert in / , F/C ia par* in A/C* th«n P in
pur* in #•
Th>er^rf1.g1)* If B i« a piir» mtRronp ©f A 8«i«li tbot
A/B iff • dir«et mm of egrelie firoup#ff thin B i«i a dirtot
ffwnnfiad of A«
PropCJiitl<m(1»22)«If H i«» o fv«¥( jroiip of G flneh tliist
tlio foeto^ i9co«p 9/E in to^>«ion'*fr«o» thon H i« ipnre in (#•
<Qttsuil(l«25)«?er • «tn\>iifrovp 5 of « «po«p A tbo
follewing oonditioBo art oooivoliMitf
(1) B in puro in A,
(2) BAII" i9 0 diroet suerend of A/hf?, for ovarj
n > 0,
(9) i f C ^ B in flttch that B/C i« f initoU eo*
gonopotodt tboa B/O is a direet fluiwand of A/0«
yhacgow(1.g4)»Tha following mr9 aoQiTalant eondltlon«
for a mtYfironp B of A,
(1) B iff Tnuro in A,
(2) P i« o dirtct ftvammA of n B, f©r »r«ry n > 0 ,
(5) Xf C la a t^eup tatv^ean S and f such that C/B is
finitaly j:;«naratad, than F in a diraet sweerend of C.
Xt i s to ta Rotad that a ^onp i s pwf in avary prcnp
obtaining i t 9^€^ly i f i t i s diTisiVla.
?haog<y(1«g5). If 0 i s a pwa sttlri pottp of tha group A,
than,
(1) ! » tr- 0 n i l %
(2) (e • i O /A* i s pora io / / / • »
(9) 0 £ ii< iBpliaa a i s i i T i s i U a .
DafioitionC 1 *26)«A sutgroup K of a gL*oup a i s oellad a
iiaat stttgrottp i f tha aquation pz « ht Htp Pt i s solvalta in H
whanarar i t ia solvaVla in Q» This i s aouiyslant to tha
raqoirsRant that,
pH • H n pG «
for e l l pe?«
l Ttry pura suVfi eup of a group ^ i s naatt^ut avery naat
imtarimv of Q naad not Va pura* ^aatna««s i s a transitiva
propirty* i t i s an induotiva t»^op«Pty, that i « , tha union
of an a<4ei9idinfl; diain of naat suVgroupe ia it^alf a neat snV«
fl^ovp. Tn torsion*f^aa icroupa naetnaaa i s acjuivA ant to
purity*
jBn| j ( l«27)« f l» lat«r«*«iiM • f t«« «r M r t mm% m(b^
grottpi 9t ft «re«9 is mit* la ff«iWfil«ftftftt« Ibtt Ck* (•} •f {%]
n^hmf A i« «f «r4«p f^hilft % is ftf ftrAcr p* fhin ^i» mtib^sttm^m
A m (ft] M i 1 « [(9ft 4 %}} «rt nftftt la 0* Sat «1M ml^gifvwp
C • i n B • (y^a) ia aat a«Bt la 9. U is %• %• aetaa tlsiel »
ia aat f«ra la Of Haat i% aMialaa y^a af hti^B^ 2«
1«28)» I f A la a fta^tp^ai^ af Ot^t» faUfla&at
atataaaita ara aqalvalaitt
(1) A i« a a«Rt aalHsreap af §t
( t ) 1 iff MKlaal AlaJalBt fipoa seat aatfraap I af §#
(9) I f f la a satffrattp af 0 aaslMil ila^alat fipaa A«
t l m A la MKtaal aiajelat fpaa Kf
(4) A / ] » • » »A, for aU ygP.
qfaa«iti«i( <*ii )> I f II la a a«at natupaay af 9 aaA
•Itbir I l t««lf 18 «i •iaaaatary t^traay «P tha faeliia graap
Q/ft i» ill«B«it«rj««haa I 1« a 41raa% •lamui af ft«
y^BaaitlMif t«lOk Sf B l8 a alalMil AlTlall^la gravy
aMtalafiui ^t B«a i la a aaat mthgp^mp af 0 I f «a4 oaAy i f
B « 9 n B lAmf B la a Airlaltta sa^sroap af B«
»gaaa«it^«i( <«11 )> I f E Md £ ara «re«Ba tlkM BMBtl}
Aasotaa Hia grai^ %t tmrnmrnf^Amm af K lata Z>«
I f I la aa laflalta israUa graayt )Ma BM^Kf]*) » £»***
X£ ^ t» a cfoUM iieo\ik^ of f in i te oz4«« a $lMa,
i i i i f Ik i s ii tocaiaa a^mi^ diMl i# ia a «Ol70la^ f2^«e
.ironic,
(2) i f ii i s a divis ioio ^i^u^ and i i s a r«auos4 ui&j v
( » i f J4 i s a i:?«>iSi7oui» and &i i s a <in < >APt 9 ^ ^
;^^«ogagiil^34i. i f ji ana M a£« ^ou^s feiie foiXoniau i^l^t
(1) iiQai«i.»i<> i s a ssrsiatt-fres i Jirou , vaondvar ii i s a
torsiiim-frso ^^«i^t
(2) i f ii i s toiroiaa-fres aad divisiS>ie» I^«aii6«ii&»i«) i s
tOj:siOAof ««e aii4 <&ivisi<u^s,
is) i f n i s divisible* tft«niitiai&t^) 1A tai.aioA-fi.oSf
(4) i f 1 i s tsx-aioa-f £<»• and divis i tas &iMa so i s
iioaiib«A«> »
($) i f & i s 6orsi<^-»fr0s aacl Jr i s divisible* thoa
iioa(iL,ii) i s divisibis*
ihaaiem^il^ iSK jUieirs oxist aa&«imL isador^/hiSMit
iioai 0 Js it ) « » iioa4iw4,a), i s f * i « l *
isi * i*i *
Qho laotiO o^ miCii hucoaocpiiiiia l a o jaX to vbe kai-iiol OA %he
aost iuxMakocpblesi*
» ^ I I I - I I I 1^ A III l>Tli i i i i i )> J • •• • » W
g • • • • i l . ^ 44 • . . • « . . ! «» U " • • • ll» W ' I ' l l • «|> \J
10
Tb« short t m e t 9i«mi»ne«t
•>A •>D • > 0
with D(MI4 hm99 !)•) d lv l»» l t l« , «hieh • x i s t t f©r •vary mrovLp
Af in oelled ma iajeetlvt r«iolution of A*
Tha nhort ccacit «»qtt«Bo«t
•>H •>A •>0
with F(«id haiiM H) ft>««t«hi^ txifftd for every group A» is
oftllod the frof or prejoetivt roeolvtloa of A*
2tfMUfi |^1.37) . If A,B.C,«i« » aro groupe and
tt* P« v> Bt M Btf hoiiMM>rphi' ei#t t *Bi tha diagreB;«i,
ara eoMcatatlTa if*
riwpaotiTaly*
fha fiva laraad.^a^.Con^ldar tha eowBUtetiva di«^P#
with axaot rewa of gronpat
A - — > A , r . > B ,
•>A.
i"5 •>y
11
(1) If 89 *"^ ^i '^* ^^^^ *"^ *9 ^" on»-^o«-OR», ttam
«« Is onto*
(2) if 02 " ^ «4 " * on»-to«oii» and a^ !• ontOt thflB
«• 1« 0B»«^O-«Il*.
0 t ¥
id eomiitatiT* mfl al l th« thraa ooltvna tra axaet.
If tlia f irst two or tha last two ro^a ara axaat* than tha
ra»ainiiif row i s also axaet.
I t ia to ta Bota« tbat ttaa 1 o n ta iroTad vithaut
A^ aad/x^ l aiBic «oiiarorphisM««
PafiaiticnC 1.40).An axtsasion M^^ff) of a group A ^J a
12
gr(Ni? B id • ptdr oeBsietinir of • i^ovp 0 ftnA a tuMrenorphisB f
stteli tliet t
0 ——>it it io f >g >o
i s m «xaet «ift0tMme»« wh«Pt i ««y »tPiid ftop th« idantitj
napping.
for any two groupa A «id B» 8st(F»A} oan t a daflntd a<i«
acmlvalanoa (dan^aa of ahort axaet saottineat
i>4 • >X >B »0
ultB addition Valng Baar addition (•aa|2|)*
l ( t ,41) . If,
0 "'>A — — » B >C >o
1« an axaet Piaevanea imd 0 1« axisr grovpt than tiM fellovlng
9aepaneaff era axaet|
0——>lteB( 0,A ) >Hcp( (J ,B ) •>Hep( 0 ,C )
—>>ttit(Q»it) >^t(Q,B) >l!»t((?»C) >0»
•Bd
0 .i-^HOKCCtO) ——>HoB<B,0} >Hew(ii,a)
*>nrt(c,o) —>ixt(B,8) —>isxt(/ ,a) — • 0 .
1?
Pofinitieirf 1.43 Ufh» •swot •i«(}ii«tio«|
0 • >A ' ^">0 *"">B )'0
in 0i l l«a fliplittiAcr •x«et i f 1 / iff A d i r to t smnetiid of a«
•na (Cl»f) i« e«ll«d a s p l i t t i s f 0xttB«ieii of i Vy B«
Bvtrj •xtiBsion of A tjr B spl i ts i f «]i4 «n l j i f
Sxt(P,J^) * 0.
fropeititie«(1«44 ) . l a Mflih of ttm fello«rl!ig GMm
Bxt<?>,it) • 0 t
( 1 ) i f A is • diYitviVlt grnsp t
( 2 ) i f 1 in • TTM JPTOUP I
( 3 ) i f # i«» p«diTiaiVlt and B i9 a p'-group*
fiftiiiyiiai(1.45).flia axaot naenaiieat
i s oalXad pvra axaet i f iA is a pura fiiiT^greop of 0 «id(G«f)
14
i« oalltd m. p v •xt«i«ion of A ty ¥•
Tb« poopB •xttnslons of A t^ ft fern e iRvVgrotip
P M « ( B , A ) of lKt(B,A) whitjk ooinoldoo with tho 1st d a
ouVgrouv of lxt(ByA)» tliAt It t
Pazt(!?,A) m r\ n 1&ct(BpA) •
io a pvr* »xeet ^Acumeo, th«i for my group a tho following
' oovoBooo oro oxoeti
0 ——»Hog(QfA>——»Boi>(Q»g) ' >Hfl»(Q,C)
——>Poxt( 0,A ) —->Port( 0,B ) ——•Poxt(0 ,C) — > 0 , OBd
0 —->HflB<C,0) ——•>ROi<B,0) — > H O B ( A , 0 )
• > Poatt( C »0 ) —>Po«t ( B ,a ) —>Poxt ( A ,0 ) — * 0.
X|LlSJtf(1.47). lot [a^ t iftx] l$o o fanlly of groopo,
thoB tlio folloilng inoKorphifwo taoldf
POXtCB, M a.) mil POZt(B«0«),
15
f«sl(B,l) • 0 •
(1) ftr a l l i t i f «aA mil^ I f > JUi « i l r M t • « • • f «r«U«
( f } ftr a l l B«if sad oalr i f A S. 9(?)it alMva 2? ia a
i i f i a iUa m^mp aaa 9 i» a Airaal aawoiA af a Uraal frataal
af ftaita «ralia graapa i
(3) i f •«• aaly i f i ie alffa%rai«lll3r aeai^al f ir a l l /B,
n»ft»nti«ili.4a^. A «re«p 0 ia aallad alfifcral^tfiily
aMipaat i f i t ia a Airaal wtamfoSi af a r « y ^ a i 9 irtdA a«i%aiM»
ia afwIvalMH %a tlui va^aiPaMBl
QBap«al i f aaa aaiy i f f a s K M ) • 0
a l l
iwyiat i f i t ia a Airaal aaa;
i t a« a vara siAgravv* n k a i i
tluit 0 is aigitoaifpsdjr QBap«4
^^l&ff Iwnalagiaa aatlMi* i t ia aat Ai f f iaat ta jfnm%
' IQMrt tte amditiaaa faxt(Vt»0) • 0 aai l i i t < M ) • < ^mVi
tkat va3rt(B«») « 0 far a i l craapa %• UNM a «raitp 0 ia a i *
gfHraifilly aMipaat i f ant M U i f i
laBt( VX,4) • 0 aai act(Q»0) • 0.
16
]}lYt9tMt upcupn wf ftlfttr«io«lljr ooepeetiH group la
elg«lir«io«ll3r «otipae(% i f ao<l enly i f itff rtdvie«d pf rt ifi
ali^atraieelly ooiip«flft*
pur* ffptgrevp la «« elfttrcioftlljr eoi paet group.
Pgaamwitioa( LSI) . I f # i» al^at^ftioally eCBpaet, than
its l lrat UlB au^grovp oeineidaa vitli i t s wnxipal dlTiniHa
auViirottp.
£CSfiSaliUB,(n52)«Xf A l* t«pfllo«-fpaa,2ait(S,A) ia
alffaVraieaily eo»pa<)t« «h{>n«v«r C ia«
l»eoa9iti«»a( 1 «g3 )>If i i0 elis^aVraiflally eowpaett than
fKt(C»ii) io a raduoad algalreioallj oovpaet grenp*
l»flt>ogition(m).If A ia a terffiea groupt then
8e«(A»C) ia a raduead algalralaally eenpaet groapffor any C.
Proooi»itioii(1.5S).A group 0 ia oallad eoternion i f a l l
of itfi axtanaion Vy teraloifftraa sreupa art aplittin^tthat
ia i f lxt<l^tt) • 0 for a l l taralon*fraa ^eupa B« Bo a ^onp
^- in oator^ion i f Hasti Q O) « 0« >/
"S^mpf algatraiaally eeapaat group ia eotoraion t
ooflTaraalyt a teraleo*firaa eotaralMi group ia alg«ftrai*
eally aoKpaat.
17
Tfcfe?tg{ 1 3$ )«lygy eotorsion is eup 0 IIM • voiqat
dvooKpesition into • dirtet wim of tlirftt ireupat
eotortion ijiroupt MA B 1A « rHuMd ooteraioa group lusTiag
no tar«io»*ftrM dlrtet vtMBasA ^ 0»
ITe»ogitiea(1»ST).i grciip iw Mt«riileii If «ii« oaly If
i t 1^ ftB •pi»«rp1ii9 lseii:« of ea ftlgtlroleolly eei»ptet urovp,
gropoiiitioa( 1 >S3).ii ro^neod eotoroion group in olr^oVroi-
eolly vmpmvt If m l only i f itn first ^ a imts/ ovip •enishoii.
Propo«itioii( 1 . ^ ^.l»t(0,A ) l« eoterntoB for o i l i roupii
l(t*<0)*Zf 0 iff « oetorslon groapt t)>«"
Rap(A«0) ! • ootoroiOQ for ooj A.
5* Kigli «st«ioieBo«
l>ofinit^oii(1>>1).Iot 9 Vo a s^m^ snd H in • sttVgr««p
of a ttfltxinel with ronpool to Rno*« 0, thin R i s oollod o
Mjfli fivVfroiip of 0«
mammmmmmm
1 M r * t1ii^ •ppiTtiBiitjr • f miptfmwtMg W9 ^—P coi^)*
• f «ratiti€« to l a t t Iff f•mm M*A» I w i s , tli« ()Bm«p BasA,
StpartMiit af itott«MitlQ9« Ali««rli V B » U » VmirmtBi^t V I M
inaii lU af Id * i i i i f f « « i t iMalHi iBtroteetA M f t h i *
Ivaacli af Alfttoa aaA •»&• « • ae^atntad wltli tha ttt« 4«r«l«|»*
•« i i « la tiM t^Mipy of atal i ta grovpa* B* ««« a «aiv«a af
graat iaa^at io t t ta »a a l l tht %iaa tbat X was aiigi«aA in
%hi9
X aa aa laaa l»diftta< ta Brafaaaav 9rat Xsltar flasaliit
BaaAfBapartBant af VatluttatiaatAlidarli INialte ttelrarsitar fue
praridlBg »a a l l Hkt aaaaaaary fa^aiUtiaa far tlila aartc* Sia
Hraaaal la1i«>aat Ja aer v«rtc \if wt^ af aaaaaragaawit IMMI baaa
• f laMBaa ralaa ta aa*
Vr tlMHiea ara alaa daa ta ay aallaagaaa la tlia 4apart*
Mat t lAa mmm^gH aa la paraalac tlda yra^aet.
I«atly X aa al^a graatftil ta ttr. latieia All IlMa* «AM
laapita af Ida \mtgt ««rtt aakatelat oaaU aiiara tha tlaa far
tjrplac rmj aaapataatlsr asr tkaaia*
S l ^ ' S a a U a M v a r a i ^ ^ ^^" lASHHOOD ) numh • 20t001«(lBiia ) lteta« Jaaa* 1982.
%mM Of oovraiTS 4MMMMMMMMMMMMMMMMk
l!MtflM* ••• ••• ••• ( i )
t* lBi«l ••wtnetii mtf •xtansieas ••• 9
9« BigH «xt«iuilMC •• • 17
1« ffeui grettp Ktxip ••• 99
9* !••% «i4 fir««ldili «x%ffifliUiit • f » tersies*
4* I««t M A 9«r««lii|^ txt«R«i«ii« ttf a ttrsiea •^ • • l^ ••• 44
2* 9«ivro«99 « d <ni«fti«it 0fmm ••• C9
1* SOM IfHHW • • • 7 i
(11)
2« Btzt * groups ••• 81
3. tf - groups . . . 85
Chapter-^ t ?tirf-Mfi^ "^^ nsat-hlgh axtgnsloas
1« Vsat-blgh txtanslons ••• 96
2* 1f*at sad purs sxtanslons ••• 102
?R ^ r # f f y
RsHolegleal alftttra in rM«nt d«ead«« hes greim
«9 ta %• • vofit isp«rt«Bt tool In th« study of dif fortnt
fftneturos* Tt « M dlMOOYtPod for t ^ f irst tin* Yy R«rri«on
i ( I that bowoloiEioal oli^olro pla^a «n i«p«*t{int rolo in ttoo
tlioery of a^olion «roiip9« Mo uw& honologiool eothodo in tlw
diooorary of grottpa of oxtoBniennt puro oxtonsions, ootoroion
(crovpo and odjuattd ^oupo* Rorriooa | 8 ) Offoln oxploitod
hOROlOffieol olgotra to inT»«tigoto tlio proportion of tho
bifflit piiro<4iigh and noat^ifli oxtfBoioR9. lAtar othMr
•atlioMitioiaao ooeh a« Paoha i 4 i t Zrola )l9it ^Ikor 127U
Talqra |89|f Haaffaoaaaor |22{, BanaVdalla I 1 i and mrnaj otliara
adoptod hoaologleal toottniqaon in tho doTOloii««eit cf tho
atolian groop thoary.
Tlio atady of th« ^etip of oxtonaiono» puro oxtaaalona*
naat oxtanaiont*, p«ra-4iiieli and naat«liifh axtaa<9lotta of a
lereop A ty a greap B dapwdo vainly on tho fronpa A and ?•
NimaFOUii propvtiaa of th^a axtanaiono, in tha aVoTO aaationad
(11)
\f obaafiBg th« f{roiip« A find B. Splitting axtvialoiit
pla^ an iBpcrtmt rol* la tb« ntudy Pt dlffartnt tgrp A
•f 0xt«islons,
Tli« Miln teplm af dlncnRflion in thia tli««i« art tha
aaat and pvra-toiich axtMiAlena and tha grenpa llazt(B»A) and
Bast (B,A) for a«Ka aballan grenpa A and B« JUttuemt
tjpaa af abaliaa groapa ara anad and bcvolegleal vatboda
md taabnioaas ara avpleyad far tha davalepnant af tlia
raanlta* Tliroagli aat thia thaala al l groapa ahieli hav»
\9m eaaffidirad T* atalian*
fhla thaais la dlvidad iata flTa ehaptara. Tba pris*
elpal pwrpaaa of tba introdaetflrj eliaptir X «B prallal*
aaria9, la ta aa^alat t)ia raadtr altli tha tarvlaolagjr aad
tha taala raaalta af atalian greap thaarj^ whloh ara oftan
uaad in tha av(b«a(|Baat ahaptara. Thia ahaptar la alaa
IntMidad ta aiira tha thaala en meh nalf-aaatalnad aa
poanltla* Hira wa hava givaa aaaa daflnltloaa and p r a p v
tlaat spealally af dlTlsltla groapat para, naat sii^groapa
( i U )
pw^^ end AMt-liigh txttntiiefis era «l«o dtfisad.
and R«xt^(ltA) \y wKeftxm tb* ^oups A end f frm tordieii
to t«r«ioB«ft»«« ftroQii** It vilX t« t^mmpr9A that i f B is
• t«r«io« group th«a th« t uMrioar of tho group Host (B,A)
io olvont tho oaoro mm tlMt of Hoxt(C,A) for ony group C.
l«t i f A i« ft tor«ioi»ofroo group oad B i9 a tornlon groupt
thai i t will %9 found thot tlio tvo irroupo Roi:t(B«A) and
Hoxt (BtA) do not l^^•t•o in a slviiar «ay« For inntaneot
Saxt( (/Z9A} iii « roduood algotraiealljr eoepeet group vhilo
Roxt^(C/S,A) • 0, flion fro lioTo fftudiod tlio groups ltoxt(BtA) p
and Rost CB,A) kooping A a toPAion«»ft*oo group ond inpooing p
no rootriotion on tht roup B* It will to oVAWTOd that tho
two group* again do not Voh vo in tho omso fashion* finally,
«o h«TO o«tani«h«d that i f Veth tho groups A «nd B aro
toritlon groupsy tho groups 'oxt(B,A) and B«Kt (B,A} Ymw
oiuilar iMTOpcrtioo*
Xn dinptar IXI« tho group Woxt(BtA) and Hast (F,A)
has YtttM wtudiod Vy Torylng tho groups A «Bd f fro« oyelio
(IT)
grmip9 9t ptiM» opa»r to dlrtet nm of eyelie group* of
i^lao mi6 vtimm pm&t ordar. ibt»et ' •(joencon oormtetinir
^9tt to novCthooTM ?.2) and r«letliig Ho«* to Bxt to T ort
(thoeriK 3*4) baro 1>o«ii o'ttotlishtd* Tmo «iit)grottp9 of
WcxtCByA), i s omo A io a tfirt«t SOP of ojrolio laroupo of
ppiw poo«r ordtr IIOTO 1^9mi oeaotntotad md tiM proportioo
of tli»i«o 9«it>grevp« and tholr ovotiMt groiip<9 hetva 'bani
fttoAlad.
fia ehaptfr IT» wo Hava eone«ntr»ta4 on finding out a
itroop 0 that la Inoverplile to tho irenp of pttra«4)ij?)s txtan*
•ilonit for niiltaVl* le ovipff A and B. ^ob grtnvn haTO V^w
navod as !!axt.fKro«pa« It haa Yoaa «ihoPB tliat an aliBiBtarjr
9-isroiip inn Haxt.-«ro«p« Alao, al l thawa crrottpa ittoona pnra-
M|^ axtwislena \y torsion flrroap^ «ra apllttlng (tluit In
H. • iroiipa) )iaT9 taan ^tndlad. <loffO of tba Important
prepartlaa of R • ifreitpii h»r9 also ^oan raoordad In thla
oliaptar*
Chaptar ? la davotad to atudsring tha pnroHsiih and
aoat'-hlgli axtan«io!i)i« Ooi^ltloaa vadv tfvhleb a f •«aat» hlerti
( T )
•xtanaion radnoM to a rfwrt*hi(^ •xtcnalon md m nmt
•xttnuion oelneld«ii with m pur» »xten«ioii hPTt YBtn dl9»
oti9ift<l» *?pllttiBff oendltloan for naat and ptvfli ifh axttn*
Aiena, oendltion on M and B tnidar vleh 1?«xt(B»A) • 0 and
Htxt (B, Ji) M 0 havt ^aan dadueod and dl9etts«>ed.
PRSIIMISARISS
Th* B«ln fNtrpoftft of this Introdvotory ehApttr i s to
r tes l l 9om9 ef tha d^flnitlotiiit F«mlt« aad other nooviattiry
"Mta of Infemotlen ivhidi wil l \i9 «««d fir^ouontly in tho «ttt*
i90i|ii«at ohoptors* fhiw i9 l oiniar don* tmly to fix up tbo tmmi»
nologf and nototienii Btod»d in tho noe[«tl and to sale* thl«i m&elt
<i«lfeont«in«d o« far oo i>o^«iVla« Tho eontonts of this ehaptw
mp9 net original and w)0«t of tha dafiniti<nia (SRd raffult* ara
tttm i 4 It! i t and I 8 U
In «iactioii 1,ao«a I'aflfie ooneapta dafinitiotKitf>l«rantary
propartiaa and raaalta* vainly of divi«titla «utitroiapiit pfura
autf^oupa and naat aiit groapa ara ii^an*
In aaation 2,varie«a typ«i ef aanet na^piaaeaa* eoti;tr«tatiYa
diai^apa, naoa«iftary t i ta of inforeation « d raeiilt«i for tha
liroiipa of axtanftion;^ end pure axtannionat threni^ hoKOlegioal Q
cathia ara ffivan, Tha important ooneapta of elgttraleally
eoKpaet and ootornien groopn aith thair al«B«tary iropartlaa
any al<*e la fe«nd in thin taoticn.
In 9«etlo]s 3 . hli?h,por«-hlgh, ntat and ntat-hif^h
•xtan^ionn era dtfln«d. ' ovt of th« known ta<9ie r«flttXtfl
vbie)) Hill I t «<«(>di In th« fiuVn»cti«nt ebeptcr^ ar« stated
vltbout proof*
1- ' evo llM>onti ]*3r CoBoapts.
DoflnltionC 1.1 ).Qroupii la wM<* ovapy olortBt hat e
finito ordar art oallad torsion ^otspa* thost in « M ^ a l l
tho olwanta U 0 aro of Inflrito ordara art oallad torsion*
Arat fifroups*
I)tflnitlQn(1.2)>A i;roup in vhicli tht ordars of tht
tltpan-to ara povars of tht aaaia prirea nuirtiar p ia oallad a
privary or pf^oup*
Propo8itloa(1.9}« tvary torsion gronp i s a diraet SUB of
pof^oops for variotta pi P,
fropoaition( 1.4)>If F^ i s tha naxisial torsion sul group
(teraion part) of B» than £/B« ia toralon-frat*
Dtfinitionf 1 »5)»0roupa in vhidi trtry tltBtnt has souara-
flraa ordar ara oallad alanantary.
Propewltif»H( 1 .<).»vary ftlaKantary p-«rrot3p i s a direct
SUV of ojrolio KTonps of tha sma ord«r p.
Propoaitioaf 1.7). i i^oup > i s alavantary i f and only i f
arery suV jroup of A isa dlraot sunaiand.
I?tfinltion(n'^).Tha i>tattini st>l>s?rottp (hid) of G in tha
Intarsaetlon of all tha vaxlval "uVarouns of 0, that I s , n c ^ ( o ) . ^i,pO
ln%9XB90tt(m oX a l l ibm oubt^oups of ^t ^^^ ^ •
»
ula faotior oX ^ an& Is afiKEio&u4 b>* ^ •
ii££iak|4SB(l«ll)» A »ab,,j:uup li o£ a tpeuu A la oailttd
essaatlal 1£ A H J | ^ g, wa&amw9£ J la « aya«»fidro eubt zroui) of A*
1 j?Q noaltlQn( 1,12 > > ^ lo olaaontaxy If and only If ^ lo
tile ooljr o&adaUal ffttO^rdui^ ia A«
tflti'^n^ttinO-^ •)- ^J A ^ > « AOellaa jraup la a«aa« a
dlrt^ot sud of inf inite oyoUo jraapo* If tnodo o^ollo ^roupa
are udnerv^od by eloi&««ite xAlc l )* tban the f ea i^xtovkj^ d wil l
be«
l a l^ *^
ittUttkliaBi^M)* A os up (# la otOled alvlaloxa If
mi m ik for a l l asii.
I t foUova fron ttie aafinl^lon «hat an eplflioc-.iilo laai^e
of a dlTlalble ;> 'aup la 41irlalble and a dlreot $iM or a dlreot
^ coduot of «j;roup8 la alvlalbl^ 11 mn^ only If aaeH ooat^oaims
la alvlalola*ji^i£the£«oro. If «^^ilcl) are dlvis ioie dab i oupa
of A, %hm ne l9 thtir mis tH^,
)^owH*lt|o«(1»l5)»1ty«ry a^nwip 0 haw • unKni* dteer;*
petition 0 • 40 ^ R , wbaro 40 i s tha Bexisel divinitl* 9ti1'«
group of Q* If 48 « 0, thon tho ircwp 'S in eellod r«duoo4.
pgopo#itio^1«t6).A 4iTifiiVlt tBtPToup D of « frronp A
i s • diroet mmmmiA of A* tbat iPi ^ • £ 0 C for «o«o «iuV»
fsroup C of A*
Propooitionf 1«17).gyory group eoa to i Voddod «« • snil-
Uproup in o diYiffiVlo foroup*
l)ofinition(1.18)>A viniftol 4iTiiiitlo group B oentoining
a iq*o«p A ifl e»llo4 tht divloit lo luill(iaj^etivo hull) of A.
Ptfiaition(1,W).A suVupOttp H of a i« oolltd a piiro
on gproup i f tlia oquotien nx « 1i,«h«t ImtHtiicN i s 9olve>lo
in H vhonoTir i t i t oolYot:ilo in G,er oonivoloiitlyt i f »
nR a R O BO
itol4ii for oi l sotwral n«ct«p« n*
I t in ooajr to varify thftt OTOTT 4iroet «}smm&fSi4 io o
pvro nvltfiFoupm 0 aad A art POPO mit^ottpfi of A« Tho tornion
port of • ttixod fppotip and it» p-oovpoaantti aro uuro 9«t«>
urovpa. Thaao fail iQ gmkxsciA, to %a dlj^aet awBRondn* It io
to %9 notod that Q « 4 (p**) )U»T9 no tmro fluV^oupa* la
t«raioQ*firac ^oupa« iit«p«t«tioii 9f pw a »»«Ttp»©U7 la a«ain
pttTO. Purity ia an induetira propartj.
fh^frmKl^^20)»l•t T^C Vt fraV^ovpn of A «raeh thet
C ^ B £ A» th«n «t b«T««
(1) i f C iff por* in B»and F in pur* in A* th«i C id
piirt In A,
(2) i f B i s pur* in /» tb«n B/C i s por« in VC»
(5) i f C i9 pert in A, F/C ia par* in A/C* th«n P in
pur* in A«
Th>or«rf1.g1)* If B i s a piir» f«it^«iip of A avoli tbet
A/B iff • dir«et mm of egrelie firouput th4B B i«i a diroot
ffwnnfiad of A«
Prepc»itl<m(1«22).If H i«» o ft«¥(^jroiip of G MK^ tliut
tlio footo^ i9co«p 9/E in to^>jion'*fr«o, thon H i« ipnre in (#•
22ltSCtt(1«^)«?or a (fntfifg'ovp % of « ^oup A tbo
follewing oonditioBo art oooivoliMitt
(1) B in puro in A,
(8) BAII" i9 a diroet suerend of A/hf?, for ovarj
n > 0,
(9) i f C ^ B in iittch that B/C i« f initaU oo*
gonaratad, tbaa B/O is a direet nuiwand of A/0«
yhtoraf(1.g*)»Tha following mra aoQiTalant eondltlon«
for a m»\0fcnp B of A,
(1) B iff TOiTB in A,
(2) P i« o dirvct fivmm&A of n B, for vrcry n > 0 ,
(5) Xf C 18 a i^otip t«t«?«0n S and A such that C/B is
f initsly j:;«n«ratad, than F i s a dlraet sweerend of C.
Xt i s to t t Rotad that e ^cnp l«t pw in avary prcnp
oontalnia/? It axsetlj i f It i s diTiffiVla.
?haog<y(1«g5). If 0 iff a pwa ttlri pottp of tha group A,
than,
(1) ! » tr- 0 n i l %
(2) (G • i O /A* la pora io / / / • »
(9) 0 £ i * iBpllaa a i s ciiTi»ina.
DafloitionC 1 *26)«A sutgroup H of a gL*oup a i s oellad a
iiaat stttgrottp i f tha aquation px « ht Htp Pt i s solvalta in H
whanarar i t ia solvaVla in a* This i s aouiyslant to tha
raqoirsRant that,
pH • H n pO ,
for e l l pe?«
l Ttry pura suVfi eup of a group ( i s naatt^ut avery aaat
sut'ifroup of 8 naad not Va pura* ^aatna<«s i s a transitiva
propirty* i t i s an induotlva t^^oparty, that i « , tha union
of an a<4eandlnfl; diain of naat suVgroupe ia it^alf a neat snr«
fl^ovp. Tn torsion-f^aa icroupa naetnaaa i s acjuivA ant to
purity*
J0ag|j(l«27)« f l» la%«r«*«iiM • f t«« «r M r t mmt mib*
grottpi 9t ft «re«9 is mit* la ff«iWfil«ftftftt« Z«t Ik* (•} •f {%}
mmf A i« t f «r4«p f^hilft % is of ftrAcr p* fhin ^i» mtib^sttm^m
A m (ft] M i 1 « [(9ft • %)} «rt nftftt la 0* Sat «1M ml^gifvwp
C • i n B • (»^a3 ift wit a«Bt la 9. Z« is %• %t aetftd tlsiel »
is aftt ffors ia Of Haat i% «Mi«la« y^a af i M i ^ 2«
1*28)• I f A ia ft ftv^tp^ai^ af OfttM faUfla&at
st«t«a<ata w t aqaivciaitt
(1) i ift ft a«Rt wm\HSt«m9 af §t
( t ) i iff MKlaal Aiajaiat fipoa ftoat aatiraap I af §#
(9) i f f ift ft satffraap af 0 aaslMil iia^aiat fipaa A*
t l m i ia MKtaal Aiajeiat fpaa Kf
(4) A / ] » • » »A, for ftU viP.
^ftt—iti«i( <*ii )> I f « ift ft aftftt i!«%Kpa«9 af 9 ftaft
ftitlMT I itftftlf is «i 9tmm%mef p^^gtmp m tte faeliia graap
Q/ft i» AtmmtWf^ifkm I i« a Airaat •uwiui af ft«
y^BttsitJMif t«lOk i f B ift ft aialMil AiTiail^la graay
aftRtftlafiui Of B«a i ia ft atftt tmhetvw^ af 0 i f fta4 oaAjr i f
B « 9 n B nibmf B is ft AiTisitta sa^sreap af B«
ffaa^jt^saf <«11 )> I f E sad £ ara «re«Ba %hm Wmit^}
AsBotaa HIS grsi^ %t immmmrfHAmm af K lata tm
I f I is aa iaflaita «roUa gttmpf tiMa Bsm^Ktl.) »l^ftaA
X£ ^ t» a cfoUM i^eovL^ of f in i te oz4«« a $IMII,
i i i i f IL i s a toiraiaa ^gsmi^ smA i# ia a «Ol70la f2 «e
(2) i f ii i s a divis ioio ^«»u^ and A* Xm &> r«auos4 ^ui&j ,
( » i f i4 io a i:?«>iSi7oui» and &i i s a <in < tAPt ? ^ ^
i?h«ogagiil^3Ai. i f ji ana 4« a£« ^ou^s &ii@ foiXonia^ miA,
(1) iiQai«i.,i<> i s a sorsiatt-fres i Jirou , vaonsvar ii i s a
torsiiim-frso ^^«i^,
(2) i f ii i s toiroiaa-fres aad divisiS>ie» I^«aii6«ii&»i«) i s
tOj:sioAof i:se and aivisiiU^s,
ii) i f 4 i s divisible* tft«niitiai&t^) lA tai.aioA-fi.oSf
(4) i f 1 i s tsx-aioa-f £<»• and divis i tas ttkmk so i s
iioaiib«A«> »
4$) i f & i s 6Oi.si^-»fr0s aacl Jr i s divisible* tebsa
iioa(iL,ii) i s divisibis*
J3lSI)XMl(i« i$) • iUieirs oxist na&urai isodor^/hisnit
i s f * i « l *
i « i ^ i* i *
9
Qho laotiO o^ oaou hucoaorpiiiixa l a o ijaX to vbe kai-iiol OA %ho
aost iuxMakocpblesi*
y • • • — • • • i > j ^ ' i i « l w « » B
l 0 •xaet Ui9a a ia a laoaaaov^iiiaa*
J • • • i . . C . n ^ ^ • • • • < • III u
i j I I I - I I I 1^ A 1 mi ulwitMJf ii '• •• •» W
l o •«4uival«a&t to ui& £aot tiiut a 1« an isostai.% Ixlao*
^ao dO&Ow ^^^i;avZio^ 9* tiMi J O£» t
g • • • • l l . ^ 44 • . . • « . . ! •!» U • • ll» W I I !• • • <^ >^
l a oalloa a &i.;.o£'& o;&iOV ao..u»iiao*
10
Tb« short txaet 9i«mi«ne«t
•>A •>D • > 0
with D(MI4 hm99 !)•) d lv l»» l t l« , «hieh • x i s t t f©r •vary mrovLp
Af in oelled ma iajeetivt r«iolution of A*
Tha nhort ccacit #»qtt«Bo«t
•>H •>A •>0
with F(«ad haiiM H) ff««t«hi^ txifftd for every group A» is
oftllod the frof or prejoetivt roeolvtloa of A*
2if|BiJiai^1,37). If A,B.C,«i« » aro groupe and
tt* P« P, Bt M ^ * honoBorphi' iswt t *s tha diagreR^t
ara eoMcatatlTa if*
riwpaotiTaly*
fha fiva laraad.^a^.Con^ldar tha eowBUtetiva di«^P#
with axaot rewa of «rotipa»
—>Aj •>A,
i"5 •>y
11
(1) If 89 *"^ ^A '^* ^^^^ *"^ *9 ^* on»- o«-OR»» tb«n
«« Is onto*
(2) i f a- " ^ '4 " * on*«to«oii» and a ! • ontOt thflB
« • 1« 0B»«^O-«Il*.
0 t ?
i« eoimiitatl'v* mfl al l tta* thraa oolwna ara axaet.
If tlia f irst two or tha last two ro^a ara axaot* than tha
ra»ainiiif row i s also axaet.
I t ia to ta Botad tbat ttaa 1 o n ta iroTad without
A^ aad/x^ l aiBic «oBarorphisM«,
PafiBiticaC 1 ,40) .AB axtansion M^^ff) of a group A T a
12
gro«? B id • pftlr oonsietinir of • i^ovp a ftnA a hovonorphiss f
stteli tliet t
0 ——>it it io f >g >0
i s m «xaet Aft0tMme»« wh«p« i ««y »tPiid ftop th« idantitj
napping.
for any two groups A «id B» 8st(F»A} oan t a daflntd a
acmlvalanoa dan^aa of ahort axaet saottnieat
.>4 • >X >B »0
ultB addition Valng Baar addition (aaalal)*
| ( t ,41) . If,
^H I. I » n n. , I II >C > 0
1« an axaet Piaevanea imd 0 1« any grovpt than Hf follovlng
9aepaneaff era axaet|
0 >lte»(OtA) >Hcp(Q,B) • •>Hop(0,C)
—>>ttit(Q»it) >^t(Q,B) >l!»t((?»C) >0»
•Bd
0 .i-^HOKCCtO) ——>HoB<B,0} >How(ii,a)
•>»rt(C,0) — m t < B , 8 ) >lS«t(/,G) — • 0 .
1?
Tlitayjp( 1 ,•?)«!.•% (Q^t Igljt** my f«s»lly of froup" ,
Ptfinitieirf 1.43 Ufh» •swot •i«(}ii«tio«|
0 • >A I • " > 0 I *"ii>B ) ' 0
in 0ill«a ftplittlAcr •x«et i f 1/ iff A dirtot smnetiid of a«
•na (Cl»f) i« e«ll«d a sp l i t t i s f 0xttB«ieii of * Vy B«
Sr«pj •xtiBsion of A ty B npiits i f «]i4 only i f
Sxt(P,J^) * 0.
3PropOititio»(1.44 ) . la ooflih of ttm fellovlng o^tum
Bxt<?>,it) • 0 t
(1) i f A i s • diYitviVlo grnsp t
(2) i f 1 in • TTM iprottp i
(3) i f # i«» p«diTiaiVlt and B i9 a p'-group*
fi4iiiy^lia|(1.45)«flio oxaot noenoiieot
0 ——>>Ji --*->0 *"»»B ——>0
i s oalXad pvro axaet i f iA i s a puro fiiiT greop of 0 «id(G«f)
14
i« oalltd m pvr« •xtwiiion of A ty ¥•
Tb« p0P» •xttnslonn of A tgr B fern e iRvVgroap
PME«(B,A) of lKt(B,A) vhifll} ooinoldoo with tho !•% UIB
outigroiip of txt(ByA)» tli»t It t
Pazt(!?,A) m r\ n 1&ct(BpA) •
S|fS£lt(1.4€)» Iff
io a pur* »xeet ^Acumeo* th«i for my group a tho following
' oovoBooo oro oxoeti
0 ——»Hog(QfA>—-»Bow(Q»B) ' >Hfl»(Q,C)
——>Poxt( 0,A ) —->Port( 0,B ) ——•Poxt(0 ,C) — > 0 , OBd
0 —->HflB(C ,0 ) ——•>ROB(B ,0 ) — > R O B ( A ,0 )
• > p»xt( c »o) —>Po«t ( B ,a) ——>poxt( A tO) ——* 0.
X|LlSJtf(1.47). lot [a^ t l 6 x ] l$o o fanlly of groopo,
thoB tlio folloilng inoKorphifwo taoldf
PoxtCB, « a.) Sfn Pozt(B«0.), 1«I * 1«I *
15
(1) f » all A, i f aiiA mOF i f B i s • iirMt ««• •f «ar U«
( f } ftr a l l B«if sad OBIJ i f A Si. 9(?) i , «IMV« B i t a
i i f i f i t l t 9P«vi» Mid 9 i» a 4irt«l smoiA af « 4ir««% fr«ta«l
• f flaita «r«li« grMpc |
(3) i f •«• iBl7 i f i is alffS^imiMllljr ««|^«I f » a l l /B,
ttaftiiltiaal1,4a^. A grevp 0 iv «iAl«4 alcifcril^tfBily
iwpiiit i f i t i t • AiJPMft MMmsadt sf iriw^rfrMp vidA o«itaiii»
i t «• A v«rt siAgreVB* m ^ i « •fsviTsloH t« tlui vt KiPMMKt
tiHit 0 is «icii^aif92dr QBap««i i f oia saiy i f fMtCM) • o «ir a l l
^ l & f f Iwnalatioa attlMia i t in M t Ai f f iwi t t« prmf
''I^Mrt tte amditioit f n l ( V t » 0 ) • 0 m i l i i t < M ) • 0 ia^ljr
that VMt(B«») « 0 far a i l craapa !• UNM a ^raap 0 ia a i *
fidraiaaily aMipaat i f aoA mVr i f I
raatc Vx,0) • 0 aai act(Q»o) • 0.
16
ClvtffiMt ircttpff art ftlfttr«io«lly ocmp»9%tm group la
elg«lir«io«ll3r «otipae(% i f ao<l ORly i t itff rtdvie«d pf rt in
ali^atraieelly oonpaflft*
pura ffptgrevp la as elfttraiealljr eoi paet group.
Pgaaoaitioa( LSI) . I f H ia al^abraiaally eespaet, than
its l lrat Vim au^grovp oaineidaa vitli i t s wnxipal dlTiniHa
auViirottp.
£BUtaaliUa.(1.5«).If A H. tiMPflio«-fi-aa,2at(S,A) ia
alffaVraieiaiy eo»pa<)t» «h{>n«v«r C ia«
l»eoa9iti«»a( 1,53 )>If i i0 elis^Vraiflally eowpaett than
fKt(C»ii) io a raduoad algalreioallj oovpaet ficraup*
l»flt>o»ition(1.14).If A ia a ter^fien group, thco
8e«(A,C) i t a raduead algalraloally eenpaet groupffor any C.
apoooi»itioii( 1.5S).Jl group 0 ia oallad eoternlon i f a l l
of itfi axtanaion Vy teraloifftraa sreupa art aplittinittthat
ia i f lxt<l^tt) • 0 for a l l toralon«fraa greupa B« Be a ^oup
0 in ootornien i f Hasti Q,0) « 0«
lT«e7 algaTraiaally eeupaoft group ia eatoraion «
oeuTaraalyt < teraieo-firaa eotaralMi group ia alg«ftrai*
eally aoKpaot.
17
T^fe?tg{ 1 9€ )«lygy eotorsion is eup 0 IIM • voiqat
dveoKpesition into • dirtet «IMI of tlirftt ireupat
«liw« X) ia • 4iTi9ll$l« ir<Mipf A A r«dtto«d teraien-CrM
eotortion ijiroupt MA B 1A « rHuMd ooteraioa group lusTiag
no tar«io»*ftrM dlrtet vtMBasA ^ 0»
Preaogjtiea( 1»ST).A group iw eot«riiloii If on* ooly If
i t 1^ ftB Apiioorpliio lseii:« of ea ftlg«lrtileftlly eei»ptet urovp,
Bropoiiitioa( 1 >S3).A ro<ii€«d eotoroion group in olr^oVroi-
eolly oovpoot If m l only i f itn first ^ a imts/ ovip •enishoii.
fropo<iitioii( 1 .S9 .l«t(0.A ) l« ooter»toB for o i l i roupii
A and C«
»rooootioa( t .<0) . l f 0 in « oetorsloii groapt thon
RaB(A«0) io ootoroiOQ for ooj A.
5* Kigll SstOBOlOBO*
Pofinit^oiifKftO.Iot 9 Vo a f^mp sna H in • sttVgr««p
of a ttttxinel with ronpool to Rno*« 0, thin R i s oollod m
high fivVfroiip of 0«
19
If H IM a Mfrto JittVf oup Of 9, tiMin X i« pttrt ia 0 and
a/H is dlYisni** »•• i U i-*
DTmitioB(1><2),If R* • 0 a ^ S i« • divi«lVl« ^oiip,
tli«fi th« txaot n«<]tttBO«t
l«v oallad « blfh Axaet <i«mi<n)M or « M ^ 9xtmnlm of R
^y S i f tCH) l9 • hlfh «nilM«roop of X.
0 --..—>A -«—>X — W — > c
ill am oittontiol axtonfilon If W iff • »titgro«|) of X and
A n tf • 0 inplioa K • 0. fha fmVgronp A in eluo eallad an
oasantial nlbfptmp of ! •
A tuNMffiorphiw f I X >T Is an t«*'>OBtlal
iMnNworphlfm i f Xar f i«i en »P<i«Btiot nvti^oiip of X,
Pofiaitien(1.64).Iat «mo»(X,T) Va tha aat of o««ontial
tiOBonorpliiinia in |{o«(l,t)« than <t1ie«(X,7) ia a aotirFoap
of mm(X.l).
Tti09yp^1><§).l>at R %a a i^eup «ltb no alOMSta of
inflnlta baiebt ( i , o H*» 0) aid S in a diviaiHa (nronp »
X9
y • M > a - «.»a —> a / a •'> u
a dlvisiiaXe i^(mp, tbmA t
irgftMMJLttailCJ»*W, i f 4*« a and «i la » toraion
y > A •" > « >J '""> g
i a a ^u£0» M^a ajc^aaalon lir aad oal^ ix' tiMsa oxia&a a
aab^aap IL aX ^ aaaft tti^i ^ ia iwiKlaal diajoiat fro» & aoA
U « i )/li ia ffwem i a tt/^
20
fh»ogt«( 1 •69)»JM% A and K \9 ftu^froops of Of If ^ i«
cexipal disjoint fi!*fl» K th«ii (A • K)A i^ pu*« in ^A ( and
htne* > lA purv^igh with rMpset to K) i f end only i f for
•11 a« K «
TJaoEJE(1.70).l»t
t e a puro-liigh oxeet stquiiieot than for any grotiif 0 tlsa
following woQttoneos lort oxaety
0 ...—^!io«(0,/) ——»>Ho8i(a,B) <——>HMK^»C)
•——>H«tt-((Jt*)—-ilHaxt J(I,B)——>Hart j a , C )——>o j, w V V
•>Ha3Ctp(C ,a)—>H««t^(B ,0 >——>H»xtp( A,0 ) — - > 0
>(1«71),Zf A i9 a torsion ^oupt than.
Bart^(A,H) mQpp f«it(A,H).
21
llaaaiKi.T«).n»» met ••<^«ie«,
0 ——>>A JL^>Q -X—>l!t ——> 0
ill eftll«4 ft ii««t axftet ii»<ia«no« i f 1^ i s ft ntftt «ii)>gra«i
of a,
Tltft tlwftBtft of tho if*eup ir«xt(BtA) «ro tho noftt oxftok
ffftQQifieoo.
yhoopf(i .7^).If tlio f»ooatBe«t
ill Bftftt oxftcfk, thou for nqr irre»i> 0 tho fellewinflr AontmeftA
• r * ftxftot I
0 >!teiKO»A) <——>ReiB(9tB) —>Ho«(O.C)
—MioHt(o,#) ——>!rojrt(OtB) —-^iroxt<a,c)—>o,
-—>i»xt(c,a) ——>!ioxt(»,o)"-—>«oxt(ii«o) — • a
MtomCT<^*74)>yho oxftet noimonoo,
0 — . > A — i - * > B
22
neat ia ViU
Um% & %B iMiyi>?ml, 44«JoiAt tsim II* tiittiijk • iM/iw i s neat
l a a/ii iaml lieoon ii Is xiQat-4%i vi^ii areo sct «o &> 1^ dod
only Iff tQt aJUL pslASs Pt
i £«imlt aaolo^ous to t^<ior«iail«7u) lOdUls ^ood in
C H A P T 1 R-II
oRoup!! or wBAf *»i> Fims-iaoH ufmnxmis of
V«ay rMidts f«r ^ e gr^anvf ^t •xtmsleos b«v«
%•#• prevti aiialogoMily for tli* greup •t pir« axtMniORi
i s I 4 I and | < U fli« «[«Mtloa8 tlmt arts* ftF« t vhst
hmppmw to tli« 9rftttlnl fftttgreoDfl of 8xt<Btii) ond F«xt(l*,A)''
So thtgr t otta t ia « <il»ilar vtDemtr? Oaa tho r«««lt9 OBI
frattini oiAigraap of lx% %>• provaA flntlogovfilj for tha
Trattiai 9it%ira»p of F«rt T Zf «0t tliea apto what art ant
«Eid vndtr «liat ooaditlotio f
Sarrlooa proTod ia } 8 | Vmt it f in m tomlon
ISPoup tliw MtixtA'BfA) i« tha Arattiai ovriraaF af P«tt(B«A),
Ali««
«oxt(£,A) . A p lxt(B,A)
(«aa ^epofiltieii 2*6}« ttiat 1« IwtC^yA) !« tha Frattini
•at KTOup of 5xt(ByA)*
H
%• iivsidiiHQr«t srmiTa* RflMlogioKl l i h t t o « i i t««lBl«Mt
art ««Mr«11jr iapiliyti t« ««r«l«|p i l l* rmnAUt*
SvlfrMV* vf «»t«r«i« cr«B9« tar* ««t la i«i«piA «•*
PcstCBfA), B ft ^Qr«iM ir««f art alwi^ «it«fsiM* fv%lMr*
iMPt «• MiiflftUsli ttet «h« frat t ia l mA^mf Bm^iCfA) u
r««dt litfUc gMt far laia l ^ t t i s i avliirvwp VcxKOyA) af
Brt(0*A)« A M ^iiwrieir of t1i9 IMtav crony* tra abiaat
•l«ilar«
Si aaatiaa f t «a arlatUsli tli« «»%«ral iaaMTfiaaw
aanatgwlBft Bast toA Mn%^ irtiidh wi l l firaipiaitly l»a «aaiai
la tka aa^f9l«
Hi fiaatiaB 9« tlia rala f l t r * * ^ I«rt(B, i ) m i
SMI* (B«l), «lMra B ia a taralaa gtvm^ tmA l^m «araia»*fr«a
25
fffovp wil l %• 4i90tti9<(9d* It Will )>• ol$n«rir«d that in this
o««« t o ^ t1i« groupflr 1l«xt(BtA) and B«xt (BtA) Ao net t aliaTa
in • 9i»iler mmy, ir«xt((y<»A) in a rtdueed algr^traioBlly
eoPTsaet <r<n}9« «hil« B»xt (VZ»A) » 0, wh«n«T«r A i9 a
tfiralonH:)raa grenp. fttrthanBora» tliair ^ahaviottra towsrda
laoporpl^iawi wi l l ta found diffarant*
In naetion 4t «a tball n tu^ tha neturafl of tha srcupa
H«xt(B»A} and Haxt (B,A)» wh«aav« Voth tha groupa T and A
ara toraioa fronpa. It wi l l ta saan that ^ t h tha geeapn
Iaxt<B«ik) And Bixt (BfA) Sahara in a sivi lar wajr. It i s provad
that i f A i s tha toraion part of A, than Kazt((/Z»A^) ia an
alKatraieally oovpaot groupt whanarar lfaxt<(/ttA) is* /laot
tha group Raxt (C/2,A^) i«i an algaVraioally oonpaet group
whaaarar Haxt.(C/ZtA) ia . P
fha raanlta of thla ohaptar era »o#tly froR \^9\*
1 • Tha Group Razt •
I t iff knoim that tha group of a l l axtanelono of A
1)7 B iff (for arVitrary groupa A aoA B) a ootorwlon group,
^ligroupa of eotoraloa(algotrai0»llj eoapaot) groups Br9 net
26
in fsmaral ooter«len («lff«Vr«ie«ll7 oonp«et)« t» Invastlcut*
in t1ii« ff»etlon tlio«« Mv grovpa of ooter«leii(alf«traie«lly
oevpaol) grovpii vhiiAi er* Rlvagrs ootor«loii(ftlg«Vrftieftll7
oovpsot )•
In t!il« AlTftetloii « • f irst prov* t t»t th* flrnt Uls
iTQt'i oiip of Set l« ootorslon.
! • ••• ( 2.1) ytxt(0»A) i s for All grcmps C ooA A •
ootor«ieii grottp.
2C2£|* Sinoo oirary group A eon %>• «Bt •d4•d on a puro
fful'lirottp in « olftVraioolljr eoepoet group B» thoro osiots
« puro osoot ftOQuanoo t
with B/A olgot^roieolly eetipool («>oo osoreiot 5 p«go 162 of
I 4 ! ) • VoVf for oay froup 0 tho soouoaeo t
BMKC,1/A) «—^Poxt(0»A) — > P o x t < C , l )
io cxoot. Tho tilgotraio oeiBpoolaon« of B/A isplioo tho
olget^roio eowpoetaofifi of Ro»(0»B/A) (soo tlioor«i 47*7 of
1 4 I).
27
(*»»» tb«»or«e 53-4 of 1 4 I)
Pext(C»A) Valng th# »pliBorphlc i^ag* of aii ©l««l:r«l»
eally coppeet aroup find hence cotor«lcn,(ai«» propotition 54 #1
of 1 4 1).//
Next, we prove that the ^ rcaip of ptire»hlf:»h extsn-
•»lon«i i^ cotorslon,
Theor'TE(?.2). If C l<» a tor»loTi jf5roi3p, then the
(JTc^p Hext (C,/) 1« cotcr«ilon far emy sartmp *.
Proof, ^inee.
Hext (e ,*) ^ Pext(C!,^),
The eitBct nenuencet
0 >Hext (C,A>—>Pext(C,A)——>Pext(C,/ )/Hext (C,*)
>0
Indoee^ the exact '^ecoeneet
HoirC C, Pext( c , / )/Hext (C, / ) ;
•>^i(c,Hext (C,/))—»lxt(C,Pext(C,A)) .
28
911IM, 0 lA • tornlen group i t f»n««« tluit th« faetor
irevp (»•• thMTflu f9.3 ef I 4 ))
?«zt<CtA)/ H«xtJCti^) F
i« r«diie«4, «liil« 0 ie • dlTifflH* crovpt b«iio« tit* flr«t
I9P0VP l9 0 • iilffe* tli« iMt groap i« 0« l!«o«ii«« P«]ct(0,A) la
• ootoraloii fsraup (laaaa 2*1 )• Coii«i««v«itljr|
lKt(0»Ht«t^(C,A)) • 0,
•BA H»Jttp(Otil) i0 ft oetorffioB grosp*//
flift ftBftloittft of thtoTMi (2*2) in ••«• of BMt •ztmaioat
i « OOBtftlBftd iB
pgono»itieB( 2.1 )> X«st(BtA) ia far a l l greupa A BBA S a
aotoraim graup*
Prea^. 9ineat
«nt(B»A) ^ lxt(B,A)
tfea faottiMieat
(>--~^f axt< B » A >----*>«r|( B »A >---»>lxt( Bf A )A«art( B , A >-----> 0
i s ign«l« flM fiMFtir cronp t
19 ywhiMi. Xi tiMlf « • • ? « iMVt %li»% i to fratt iai IM(1»«
iEr««9 i« 0« flM ••Ml ••^vaae* imimttii Hw «xael t«^MiM
•i-«»>lir^ Of ttc^ Bf A ) )•
•B4 tlw fibet«r gr««p i s a r«d«Md «r0iip» iil««tXKt(B«i} i« ft
oot«r«i«i 9tcmp tm e l l grotys i laA B (•#• tlitirMt 94«i of
I 4 )}• HMO* HI* l««t group TftalfflMt* n«r«iMr«,
Mii Y«xl(B,i) i« ft ftfttftrsioB 0n»«p«^
la the ii«cl 9PftBftftiti«i ftt Almmmm waAm «liftt oftiiAiU«ift
wi l l lftxt.(OtA) !»• ftlcfttrftiftftlljr «M ftftl«
JBBtlli|||fl|^*4)« Zf 0 ift ft t«r«iiB fffftftpf tlun
S«K% (0»Ji) ift ftlgft^ftieaijr QMipftel, «li«i«v«f f«ct(CtA) i* .
JE|f§£, IMliV til* gl^in QOBditiftfts i»ft l» tiftii ftf
30
what w« hMW iNMr«d in lMHift(2*1) P«xt(CtJl) its « r«dite«d
%oterf9ieii greup, vMeh It •IgcVrml^lly oo«p»0t also. Ranot
tha fir«t tJla fvuVgrevp of Paxt(^tA) «no^ vaniali (aaa tlia
ravark folloving oorellary 56*2 of { 4 !)• But i f tha firnt
VI* autgroBp of PaxtCCf/) voaiahaa* so doaff tha fir^t Vim
maitfefevp of Haxt.(Ctil)* F
Thus tha f irst Ul« su^grovp of a radsead eoteralAn
airtmp Haxt (C,A) i s 0« Ranoa ( t j proposition $4*2 of | 4 ))
R«it <e,A) i s ali^alraioally oo«paot«//
AB aooloiEOtts ra««lt for naat axtansioas oan Va proratf
azf>etly aloaf tha sasia llaas and i s oontainad in
froooai^^pf(g.S), If %ct(OtA) i s algaVraioally ooapaet*
so i s Vsxt(C»A).y^
CoBoirBiiii; tha faetor f roupt »a proTs tha following
ftropositio«(2.<). If 0 i s a toP'^lon group than tha
faetor groupt
Paxt(C,A)/RaxtJ C »A) p
i s algal-raie^lly oovpaot for a l l groups A«
31
froof. fh9 frftttial nu^groupt And b«iie» th« flr«t UIB
futupreiif of t1i« factor la onpi
Poxt(C,A)/8M(%^(0«A)
i t 0. rurthonrorof tho ho«o»orpMo imttg9 of • oetor^lon grevp
i« ootornloB* It fell0»« tbot tho faietor group t
?ox%(C,ik)/RoxtJO,ii) P
1« • roaveod eoter<«loti froup who^o f imt tll» oiit grottp vfmlshoff,
Rone* It iaalgotriloiilr opiipnot.//
In oaao of N«xt» tlio proof of tho fellonlaf propooltlon
iff fliloor. •
2S2B8allia]^2.7). Tht fftoter groups
lx%(0»A)/lioxt(C,A)
iff for a l l groiipa M aad C aa algabraioally oespftet group*//
a- Tha Qroup Vast.
In thiff aaetloB, va shall proTO nona Inpertant rasulta
of tba group of uaat axtflnaloiHi, vhloli wi l l t o franuimtly w»ad
5t
i» tlM nans A • ViMtr MiftloiiVi far thB m^n^ •f fwr^Hbi^
«ct«i«i«M wilt ^ •%«%«<• Vk* 9r9«fli» i«IUl«ii art « !««
stailar Uat» •• tlMiic of nm% •XUBSIMW «P» iBitt«A*
fiMt «• proT* ttet Vwt(0»A) i s ^M llf«t%i»t MtgrMiy
• f l i l (e ,A).
JtBttt* Vte pMMf Af this propMitiffi iM m mmi • ••»••
«««ie» •f t]iior«B(53«5} • f I 4 i • «liiA •%*%«•» ^WKI M
•sttMiM a of A ttr c it divisibia 17 « viPiM t if wa Mijr if j /p i i« II airtet immmA. %t O/v *
Art, i/pA \wlm% % 4iy«o« • i i t t •f 0 / ^ ftor a l l v«y i s
•«Uv«l«i% %• «k« M M B M * of i in 0,
Aifl«CM •f tids vr«pMiU«ii tn tlui grMp vf pv«-ldi^
«Kt«Miai i« ooa««iii«i tBf
n i«xiy(e»i) • y fPcrt(e,A)*
53
Proof, fw iroof of thl« proposition •^••(thoorof! 7 of
I 9 1).//
7a th* n»xt thoortip «« di«iaanf9 tbo naturBl inoKorphifn^s
of tho ti oapii of noet oxttiK^lons.
Thocrg»(2.10). I*t [a^i i « I j to a faeily of irroopw,
th«n for any powp H tho folloving l«oifiornhi«B?i hold*
Noxt( ® <3.,H) Siw ffoxt(a,,H),
m * i i *
1ftxt(H, « S-) »« ??txt(H,0. ).
Proof* ?r«ttinl «?ut!?roui>« of ttw© i«€i«oPDhie ^roupn «ro
i«c«!crDt;iCt and FlrettJnl «ul#?rottn of e direct iwodwct i«»
tho direet Turodttot of tho frottini aul^i^oups.
Tho i9oeQrpl)i«ii of tlioore»(5?«2) of j 4 it
ifel * ifcl *
DpCSJttC ® 0-,H))af DpCit B3tt(0-,H)), pitfP it! * PfcP iti *
PISP ie l * i«I p«P *
54
i*9» tM*epo!iitlen(2.9)«
M«ln« th« iieoond isoteorphisn of th»or0s<53«2) of
I 4 t, that l9t
1«I * l T *
p P HI * ptP HI *
pel* HI HI p P
HI * HI *
Xn m»9 of pur«*-hl|i 9xtminlon9 th» following;
3ai2SEfl|(*-^^)« !•* [^t I 4 t l } *» e fiwlly of
to-^lon (vmpwt tlioti for any tor«lon ^oup H tb» followiiif
ifltoiPorphlfiiR* holdfl p^pe^f
«•«*„( ® 0,»H)Sn Ho*t (0,»R), ^ HT * HI ^ ^
P HI ^ HI ' ^
55
2KJSS1* «1J>»« iropo«itloii (2«9), th« proof of thin
lirepooltloii ! • mob tlio omo M tliot of theorw(2.10) ond i s
thoFoforo o«lttod,|(^
9« Vont AiA Poro«fli«li IsttBoieno of A Tor«loB«>firoo
droop*
Tho oiii of thin sftotloB i^ to mt»0T tbo aoot ana p«ro*
blfi i oxtOBoioiMi of 0 tor«lGB-»froo groop A t j any sroup B.
flBot vo fix tho grottp B • Q St «BA iirvofttigf!t» tho l abf vieur
of tho groBpo V«xt(B»A) and Roxt fB,A)« Vrntt^tf ivtodj tho P
•xtonnioaa for any group B* Tho ronolto wi l l Y'O prorod for
aoot oxtonoiooA and thoir analoipoii for piiro«4iiffli oxtonsiono
wil l l»o proYod i f thor f »o^ eloBf oivilar linoo.
In tho foUowififf thoor«ff wo proro that 9o3rt(^Z,A) io «
alf^ol'raioally eeapaot groop.
21ttSCM(2«^2). lot S ho tho divioiUo hull of aagr toraion*
frao srevp A« thoo for WXT nonoBorphifia g svdh that
peP
vo herot
llaxt<C/i;,A) Sllo«(C/2,(l)®ii ( A / P A ) ) A A ) . p«p
• •BMy
9«xt(€/Z«ik) i s • r»<!iie«A nlgvlTftieBlly oovpeet group.
»—t, 9lBM» I> 1« tiM dlTi« i l l« talll Of I , t)M
««4MiB0*t
0 ——>>il ^••>P — > V A — - . > 0
i« an •xaet ••cRi«iie«* I)«fiii« « »oiio«opplili»« gt
la viioh a vagr that «
!<•) • (f(a), {• • pi} ) , for a A .
firat «o proTo ttiot gA la a ntat ««>group of
1>®« (VpA). MP
If 9 ia tmf priva B«il)ar and a e Ft *p « A
aad
< • # ( » . • pi) ) e » ® « (A/pA), •' piP
«t«eli tliatf q( a , {"p • PA) ) • g(a)
• (fta) , (a • PA}) ,
37
tbast
0 • • f(ft) •
« {% •pA] - (a • » ! ) .
Til* i M t •<niftlit7 mspllM VaiBt t
3«VPM«9
« •()•• t for »*e A,
TlMB»
- 4(«(«*)) < q(«A).
This prcfV9n that M 1» » n««t «9uVirroiip of
»(^!i (A/pA).
ffe«p«fiiff«t ^ « tftfii«iet I
iM mtftt •mi* tnA yittd* tli* «ia«% ••<iiiiie« I
^ • « i
fkiff first 8WMMI4 i« Of mimm t i t a t«pci«i frov^
6Bd B» tli» 4iiri«iU« luai •f • t«rci«i-4)rM irmip i» t«rci«i*
• « lM<<yS,2(p))»
m 0.
9iB«*» C S is • «iTl9iU« ir««f iB4 8(9}t tlM ^ ta iU
idlM tlM last «rM9 la tht ^it^MUM (1) f
TiM t i ra t mi«Hi4 i s O9 t iast S is s i iv is i^ ls ffpssp.
Alss lor «li«rta<t»10) vs lw?s §
lMEt((/8»s ( V f A ) ) ais V«X«(C^StV9i)»
• % IS««(</S»S(9)),
• 0 •
9iast« JKv) is sa AmmUafj p" y sap sss v«MHrlc($*7)<
ptP
fhas tlw first sad «hs Isst srsn^ ia tbs sxssi
ss«i«ies (1) Taaish »
40
Can«»qtt«ntlyt i** ottaint
!f txt( t/E ,A ) S Ro«( C/f ,(1) (Ti « ( A/pA) )/ff A ) . pel'
''Inet c/2: 1« « ta"«»loii «frcui> I t fellows *roff th#or«r
(46.1) of i 4 t thett
oftd !\»n«« 9«xt(C/?*,A) l«t a rwluecd al^«traietJly eospact
now tocvt»in€ on par«-5hlgh txtmsloRti, wt oVtaln fm
iiit»r««ting r»<«tilt« that 1» i th» DuroHilffh •xt*r3«lon«i of
« tor9leB«>froo ??roup Vy C/Z ••rllt'^ end !•? contalnad 1^,
,2.13)• ^ r a ter«»ion-fr#o saroup At
H»xt (C/2,#) • 0.
Proof. Ty propo<iitloa(2*9)t
^ Pt;?
'^iBct, ^ ? Iw II tor«»loii ffro«p i t followf* thct
T»txt(C/2,A) In a rodvetd rroup.
•1
On tb« oth» ImA (ty prop«rtr * P««* 224 of I 4 i)
iext(C/Z,il) 1« an ftl^arrpioftlly owppaot group and Its firnt
mm ffvtiroiip P»xt^«,A)wwt to dlvifiit^lo(«»o «x«rel»o 7
PMO 1€2 of I 4 I)
Hofieo, VBxt(Q/%9A) oad thoroforo
p
CoreU»y(2«l4)« for • torolon group C ond to] <«ioa*
fipoo group At Pntt(C,A) - 0.
In tho aoxt thoorap »• oxarine tlia groupn 9axt(Cyi)
and H«xt <0«A)t «haii C la not raatrlotad to a torsion
groupt ymt A iff a toraion^fraa group.
yhaopa«(2»1§). If A in a tornion*fi*aa groupt than
for any group 0» tba following iaoBorphiaB holds goodf
irait<CtA)«Vaxt(0^,#) 0Haait(O/O^,A),
whmP9 0^ ia th» torsion pert of ttaa group C.
iSSSt* ^ivoa 0^, tha torolon part of tha group C, i s
a naat "ut rroup of C, tha following aamiinoai
>0 »C —>o/C^——>0
42
i« BMt nao l and jUlHw th« •xaet ««qttflBe«|
HMKO^VA) •• •»irtftt(C/C^,A) ——>I«tt(C,A)
>lf»3ct(C tA) >0,
til* flret group in 0* •inot 0^ i s • tortmon group and
Af a t«r«loa*fr«o «ro«p«Jil«09 in tha naeond group C/C^ la a
torsloB^^aa ffroop end (trm axeiepla 13 PBga 226 of | 4 | ) |
p lrt<C/C^,A)- fttt<C/C^,A),
for a l l ptP, Hanoa fifo* propoaltloB (2«8)|
»axt(C/C^,A) - p^p P fct(0/C^A),
. lrt(<C/C^,A).
Tharafora* ir«xt(C/e^»A) in a diTlsit la group, hanoa tha
axaot aaquanoat
0—Mlaxt(C/C^»A )—->»aJtt<C tA y—^!laxt(C^,A
i* a fiplittiag axaet 9a<m«noa*
Oonttaouantly, «a barai
Haxt(C,A);i laxt<0/C^,A) 0Htxt(C^tA),//
43
m 0Mi« of oiir«<"hli^ •ztfBfiioii »• bar* th« followinn
ttnalogttt of thtor«B(2»19)
f1ioflfa«{2.1<). If A i9 a t<iF<9loii«*fraa proup, thflB for
aiqr group Ot tbo follovlag i90vorphl«Bi holds goodi
mhwf O t i" tM torsloD part of tha ^OIIT> C*
Proof, ftroeaadlng along tho ffwa linan an in thaoraB
(2,15) and lATlaa of thaoraa(2*13) tba ramiirad InororTtbltinR
holds itood*//
for a tor«loii*f!rao irrottf / wa astatllsh tha following
gaDarali^otloQ of thaori«(2«12)*
gropo«itioii(2»17). If A la a tornioii-fraa group» than
iraxt(0«A) la an algatraloalljr oevpaot gpmxp for a l l groups C«
froof* tha para axaotna^s of tha saouaiioat
0 >o^—'X? —->c/C^——>0
Indvoas tha axaetaa^s of tha •racwaneai
Paxt( C/C^ , A >——> Part< 0 , A >-—*>Paxt( C ,> )-—«^0.
44
fh» lm% 9PM9 i t Otiw mnVlmpf 2«l4}*A9«fC/i3^ U m
tmnUm^^m ge«m» n 4 linpAi«« 7«st(C/B^^} «iA lutD«t
F«xt(e,A) im ft AiTi«iM« 0p«ii9* 9ta«« wa •pS»«ppliio ia»g9 of
« AiTi9i%l« isTMiv i« AiTi«iU« «i« ft itviftiUft griiip la ft
Alrft«% ftWHMBA ftf nvftiT gir«tt9 IB iMleii i t i * «tiat«Uiftd«
tiMrftDirft « t iMVfti
fto c i^ ma fftelar ftf ait(c»A) is eifi^ftiftaUjr
0mpmt^t ftiaoft Vim fmtrn^ •t 9i^m0jm m^mfm ert ftlgft^aul*
mHVr ftMpMrl ( • • • U M V M I 94*5 • f I 4 | } * But • srw«i»» ie
ftlgilfcvftiMiily ftifft#» •xftolljr i f i t « ir«iiftftt p i r t i » ftlgtl-vfti^
tfUlf ««ifft«l« n liillftvft ^uit tetCOtA) is lAgiirftiftftlljr
OMifftftt* ftpftpftftiUftS (t«$) ivpliftft tlMkt 9«it(C,it} is slgiJtofti^
Sidly s M i ^ s l * /
4 * Is«% i a i fttrs-4U«li i^EtsasisM Of A ^eersisn
Xh this «s«liw «s slMll fftttdlar tiM ssst aid y«rs«»
b i i ^ sit«si««8 of ft tsrsies «r««p A tgr « tspfilsn srsvp B«
t« f is tiM tsrslsn grosp B » C^» «fi6 Iirrs9tiirftts « M I rsls
plaQTsA %ar Vsx«(<^2tA) ast ItaEl Cq/2,A) Har ft tsrsiea «ps«p A«
S3
In ttw fellowlBi tH«orffi «» d&sews tb» \!»hanrloiir of
«ii« f!r«ttiiii cHl irevp of tttCC/ZyA).
y^«orf(2.19)* Xtt 0^ t$« tH* tertioa p«rt of 0, thaet
Effieot H«xt((^Z«<l^} i« an algol^roieftlly ooapAot group«
UllOBOTtr, K0Xt<(^S»0) i s .
groof. nne«» tlio tornion port 0^ of tlio group 0 i s o
ttoot •ttt'grottp of df tbt «o<ni«B00t(«itli tho nototion 0/(^ • r)
0 >a^ >q » >o
io soot oxool wa& yiolds tho oxaofe «oir««iioot
Io»(^«, f >——>»oxt(C^,0^) -—>!loxt(C/2,ft)
'" ",. »ytfEt( c 2 ,f ) >0.
fho firot group io 0* olnoo C/S i« o toroios ^roup
•ad r, ft %oroioii*froo grovp.
lot S bo tbo divinilslo hull of FtthtB tho oo^oaoof
0 .M >fi i»,ii>p "MP/f — > o
io oxoot.
4(
g , F - ^ >|)^K (f/pF)
(^(•t «« ia tta9 proof of tlioQr«i(2«12))
Voir, hj th««riB(2.l2) and (ixenplo 2 pei$t 49 of { 4 ))
« 0 bOTOt
»OXt(C/«,f) S!fei<Q^«,(B®« ( f /pF)) / f f ) , P«1?
£ naB( ® t<p-),(B(J)n (P/pF))/fF), peF p«F
ifn Bo^«(p-).<D(g)ii (F/pF))/gF). PiF pi(P
Butt l oooiifft of propo»ltloa(44«3) of ) 4 U tlio products
oro tor»ioiiHbroo, honeo R«xt<(yEtF) wtagro tor«loB*f^M,
F^Mftharworo, t^ prope«itioR<2«5) 9oxt(Ct/Z9&^) !« o
ootorolon group and so tho sooufBOOf
>>Vozt( </2 .d^ >• ->W»^ C Z ,0) >yoxt( q/fe »F >—10
47
•plit«« and «• h«T«
^ine«, vvwy dlr«et supiMBd 19 a piira autgproup i f follewa
that tha aplittlBf saoaanoa i« piira axaet*
AlflOt a diraet mnrvaad of an algatraleally ooKpaat group
la aliratraioally oovpac^ koiea I t folleva that ]faxt(C/Z«0^)
la alga^raloally ooppaet* vhanavw llo3rt( C^ tO) la elstatrel-
mllj ooBpaat*//
In oaaa of piira«4ilitli &x^m»iomm t)iaoraB(2«19) takaa tha
felloaliig fOTK*
Thaera«(g.19). If 0^ l9 tha twnlen part of 0, thflnr
Haxt^CC/8,0) • Kaxt^(0/Z»&^),
£[SSJ[« Tha aaeuanoat
0 "">a -'Ml »o/tt^ >o
la pura axoot and indaeafli for tha i^oap Q/t «ia axaet
aaijiiMieai
BQ«( Q/f ,0/0^ ) >Pa«t( E ,0^) >Pa«t( C/E ,0 )
——>Paart( 0/2 ,0/0^ )——>0,
48
Sine*, c/Z iff ft tertion group and O/d^t * tor«loii*frM
iroiipt th* f lmt aiifl iMt frdup* in tiM •xaet 8«9t«Bet
ar« Of • • • (M>rol]jiry(2*14) MA tli« «t<|B«ne«
SlBMt Arattini sut^roiips of laoaorphie greiip«i aro
iaosorplile i t follows Imrioii of poopooitionC 2*9) thot
:U!% «s iBTOotigato tlio l&<li»vlo«r of V«xt(B,A) oad
Host (1»A)« If A i« not ro9triotod to m tornioii group t^t B
19 a torsion group. la tb l t 4[lroetloB(oxirel80 11 pago 231
of I 4 I) provldoo tiMit If A la SB algt^^ioally eoapaot
group «Bd 0^ tko torwloB pert of C« thon tlia oatural iooBor-
plilflai
lKt(C^,A) £|]et(C,A),
holds good.
If w« cofiewntreitii on nmt md fiwref-hl^h iP( t»n«iou«i
is» e^jnarvt that analoeou^i lfic»sorTihl*»iss hold irood for © oo-
toar^lon aroti^.
I»Qpo^itloii(2.20). For « eetorsloii s!roi»|» #» th»
follQwlnis: i offiflarptoinflrii holdf
H«xt^( C^,/) • H«irt ( C • / ) .
Proof* Tht n««it »xact 't^roaficof
indttoe!» tta« oxaet «tott«ne«i
If OxtC C/C , A ) >!l «irt( C « A) >H03lt( C , A ) •••> 0
<9iiioft» A i s « eotomlon group, th» flr«t ffroup veniflihoii
O >Wwt( C ,A ) >?loxt(C^,A ) '>0
Iff OXACt •
Tho proof of p?ooond l«ioirorphlfw in r>vit« cl»«jr,//
movp Of VU9 mTwnms of CTCIIS OHOUJ*
Hsrrlfiont Fiidi««»RaiiJ7M«am't1^«n«t€iill6h>B«n|>tfl»l90h and
«8tQr ethart in { 6 I,I 8 | , i 4 U( 32i»i 29UI 1 it i 9 ( and
1 101 tte*dlfi««9fitd th« gr«vpa of laitaiieloii^* !««• •xtM^lofifl,
n«8t 0zttfi«ieii«« hli^ «5rt«iiiilett«, par#<-iil|i(}i •xt«n9ioit«i and
B«at«liii;!^ •xtAfmloRS of • sTOup I ty • group B* Variy r»*ulto
««ro •«%t«Vll««h«<) and tHa tahavievr of tha alora irmitleiiad
<!ro«9n aa« uttidiad t y el)«i«iiig tlia grevpa A and B.
^9 vlali to study tba naat axtioslea^ of a group A \y B
group f^rmjiAg tha grocip9 A and F, Rawtrloting tlio isroiipfi
A «Bd B to eyelle greapa of prlvo wdar» diraet imm of oyelle
jBToupa of priea ordtrot and direct AUK of eyelle jreapa of
prlna poaar or dart # «a nhall invo'ttigata tlia rola ploy ad ty
tba iToap of aaat axtanalonff*
t t follewfl froK rMiarlr(5*7) In ehaptor T»tfiat i f / ia
a eyelle f^oitp of ppina order pt ttiao Waxt(B»/} • 0 for all
51
greniw B. AlffOt V«n(B«A) m 0 It f in ft egrolie grovp of pri*«
erdar p for • vw j fr««P A «•• propenitionCS.f) in eli«pt«r • •
^ tli«orMi<2*10) In ^apt^r I t i t tellmn tliet lf«rt(p9A) dO,
i f B in t i l * dirtet mni of egrelie ^o«r)9 of prist orders.
In thi« diaptffr «• isihall AifiCKiii* tht <!iherftet«rift«tion
of tbo group !rMt(B»A), i f A i« tho diroet mm of ojelie
grenpn of priao or prist povar ordar and nhdll invosti^^to
ti l* toh«rio«r of tlto s^enp llaxtCBtA}^ i f ? is tho dir«ot 8t»
of egrolie groups of privt powar ordari*• fliroiigboiit tliin
flihaptar liOBologioal vahods ara unad^oxaet naqiiancas ara
oftatliRltad to davalop tha ranu l^ .
In saotiOB IfWO disoofin en M»et sofuanea eonnoeting
llaxt to no» tlkaoraR(3«2)tii«ing tlia divisi t la part of a groop
0 and tha diraet wvm of cgrelio grottpir of priso «rdar^* }faxt»
»a antatli?fh an asset ^aooanco raletiag Ros to net to Haxt
ssiag tha asset saooeneat
pitP PttP peP
urith tha halp of ajeset safVMieos of naetion 1 ,
92
« • ptww9 in fiMtioii tf tlwt %h.9f •xist tvo 8ist»ir0ii9S «f i n *
STMii Wast, ea« aoatsltitd la tli« «tli«r m^ dit«M9 the
89M la tiM«p«i (5*1) tlwt ««• of tiM ftittliHit ggmvf i»
fit ••«tl«n 3# « • eoepi%« the gr«ii|^ Vffiit(Bti)t I f A
t« tSMi%9 9RA iBfiaito criaiio fPMiF tmi, B is • lorelle grcttp
of prla* 9«r«r
flMi r«Nilt« of tlilfi c ^ ^ c r Art fHm (19 i v«A I t l t *
Bnvioiii 4i«ciMi(ioa In | 7 | m mtmtt 9«^iiiet rolaUsf
i^xt Md B0» and ^r«T«i tha ffAl««lA« tuaaraii*
19«1}* I f A la a iri*«&P tfiilKMit ali«aa%a af
IsflJtlta liai4M» iH^t ia i*« 0 tteaa tlui «tfaciaa i
0 •—>f<aji^e^tA) «—;»»igiet(0«A) • >ltea<C^,V^>-—»0
iii aaeftet. fiiart 6^ ia tlM a^" UUi tmt^m o^ c^ that la i
C • 0^/d M t A |M tha X* «Ale aMUf latliSKii af A, snalai that
tlia na^aaeai
.>A ««»>A <-^>ji/A -—> 0
93
111 puf «K8et « l th A/A dlTl f i i l l t .
Preef. *»•• Hiaprlsco | 7 U / / •
/Bftlogeiw to th« txaet «eciii»ne« of tli«oraii<3*^) «o
proTO in thin noetlon on oxoet stouoneo oonnootlng Roxt
find Hen, with tlio li«lp of tho 4iYl»illo part of * group
0 ond tho dirtet ff« of ogrolio ^oupa of prlso erdflr.
Thoyiy(l«2)#If t in th» divlfilVlo pej*t of « isroup
Gf thon tho mmet sooutiieot
"Xl/I)
3rl»ld« tlio axoot «9ociionott
0—>»«Kt(&/P, ® a(p)) ——»r»oxt(&, ® Z(p))
pitP p«iP
vhoro S(p) fitanda for ctftHie group of <xrd«p pi
PgQOf. Tito OXaOl ffOi^QltlCOt
0 — > D — - > a — - x j / p ——> 0
94
Induettt tb« «saot •I«Q««B€•(«•• thaor^^ 44*4 of} 4 i ) t
->Ro«(P, « Z<9))«
But ill* l a s t ffrwipt
91no« S in • dlvlfiiVl* (ireiip ^nd SCp}« a fiorelle group
of prlvo ord«r, io rodpotd.
wo obtain tho oxaet soouotioof
( t ) 0 —>Hoii(a/D,ii 2(p)) «——>H0B(0,it 2(p)) ——>0
tHiP PiP
frow thoeriii(9«2) and oxireiao 9«14 of { 24 } vo
knew that ® Z(p) oolnoideo with tho tioslRol tort ion opl^irovp of « S(p) OBd tho f a e t ^ groppt
WP
peP peP
i« diTi«il^lo. It follows ftm tlioor«ii(44*9) of 1 4 I
55
that tbt 9»«iii«ae«|
0 — > ! l o « ( 0 / D , « « (p) / (i) SB(p))
^^^ >Ho»<0, « B ( p ) / ® Z ( F ) )
i« •xsel«
Tlw MotlMa tflrsien aiftgroBp 0 2(p) of % Z(p} !« pgP p«iP
ntftt la « Z(p) MiA thtr«f«rt \\m Mt^meaf P«P
0 > (^ %{p) ——> « 2(9) — - > « 1(91/ ® Z ( p > — > 0 piP peP PIP piiP
HeiKo/Df « «(p))—>K«iKO/l>. « Z(p)/(5) z<P)) piP peP p<»P
>S*xt(0/l}9 (±) «(p)) .» .Ml«3it(0/ l ) , II Z(p)) , pi» Pfci»
H«B(a, « z(p)) >RMKCr« It 2 ( p ) / ® e(p>) piP pep p«p
>>8«xt(0, 0 S(9)) -—->P«it(G« n Z(p))« p*ip p^p
56
^Ine* t
Also*
%••«•«• Z(p) ! • •» •iMCBtwy p* gre«9 «•• rtvarlc 9.7 in
t« oVtftln tb* •met fi«ott«ae«9|
HOB(a/B, « »(p)) -.~>R«i<0/X}, % 2(y)/( i ) E(p))
(3) >VtKt(0/X}, ® «(p)) >0. piP
n4
(4) IIOB(0,II 2 (P) )—•>H«i<0 .« « ( p ) / ® 2<p)) PSP PiP pfeP
P8P
T1i« vbttirt «Kiie% ^tmmatm (1)«(2)»(9) aad (4)
yl«ld tb« fellovisg eMevutfttlT* dingraBf
57
0 ^>Hoiri(a/t,it Z( X,) )—i->Hotl( G ,1? Z( p ) ) —-XO
• 1 N ^ N § 2
•>HOIB( a/1 ,1? ' / (p ) / ® f ( p)) —2->KOIB( a,ti 5i( p ) / ® » ( p ) )
^
i ptF
® 5^(p)
^
"^Inea,
N » x t ( O , 0 2 ( p ) ) end fk)K(l), n E ( p ) / 0 E(p)) PfcP PfeP PfeP
lialnsr •t»lir<5Pphle iiiosftB of t
pt?* PfeP
v l t h ir«rn«ln fWiSfg nnfl TiBfg •
/ l « O t ^9 h«V»f
Twir • Iptg'^ " lB f;j«r- i Itrf,
59
tlM tliird r«» 9ftii \9 9xtmAaA to i
•>HoiKlJ.« HP)/ ® 2(p))
ifov th* thr«* eeli»m« and f lmt two rovs in tlio
eoiriiQtatlTO dla#r«i mro oxaot* It felleirs \j 3 K3 l«f8»«
that tbo tliird rem in oxaet.
wa at ta in tl&o axaet «acrttaTHoat
0 —>iia3rt(0/», ($) g(p)) -~MfaKt<9, ® lip)) p%f p^f
>SOii(l>»« l(p)/ ($) 2(p)) ——>0 PfiP P< P
aw daairad*//
An analofOtta axaet «i»e«»nea for piffa«lii|^ axta!i9io?!M
irhiOh ean l a proTod along tlia nama Unit ia a» follwa.
yhaeyif(3#1)>If D i« ^ a dinaiVla part of 0 tba axaet
ftaooaaeat
indveaa tlta axaot 9»m9n^ t
99
IB tlM n#st «li«crOT!» « • eo^cet Bfln to fxt t o W»x«
in t ^ teem of an «itftet II«<IO«TIO«
yiitwptff(5»»)*f!»» •ii»<rt -i»«fB«a©«>i
0 ——> 0 a(i>"" ) -X-> (^ i5(p») -.I^> 0 7(1,) —.>o PifcP WtF PftP
C94 iwawwrphiwit
f I SjEt(&. 0 X{p®*^)) — > « K t ( o , ® SKCp**))
ispljr tdftt
l » f . i » » i t ( 0 , ® !!(p»)) P«?
pep pgy
->RCIR(0» ® 2(p))
<0
i» txft*! fUr •Tvjr «r««p 0» uliar* 2(9*) tUMs f ^ «Qr^i«
—>f — > » — > 0
liwKKf® «<f**^»—*-»ftn(0, ® «(|P*^»-—>lEt(f»®Kf?**^l yt? VKF 9CF
"1 ^
U
pfF 98' Vil^
8«l»9roit9 of • Aftr«e% M B (6tr««t 9nMhi«l} i« %hB Alrt«t
<l
Firaniiil tnitgr«W9» of tpo iiiaiiiarpliie ironiMi » • umwpUe,
*• !»•• for a 2 t »
Tli« grmtp R i9 ft M ana h«tio«i
H«i<B. ® «(p")) Sue ® I5<|»")) .
J^p 9 »«^H» ® «(p"»iir-() p i( (5) E(p»)). r \ 3 / - ^ r # # « M^B WP
• « i41»p( ® Kp»)),
p«p
62
Rme» f« stands for th* lne1ii«loii nsspplng, end
Ipf* l* th# frattmi wl srroiip of Ho«<H, (J) Z(p*)), wMch
1«< 99Tip^ imdir tht het«o»orphl)»« « into tho ?¥«ttinl «»uV-
c^oup of is*t(a, (•) ^<p'*))« n«io w« hevof
If»f» • Imff^h. m Jwk ti,
«o iBlcf \ «f*d h»tic» Ipf» ifi ttftppod Into tht ^ftt t ls l
?iti1?«rrettp of S»t((», (•) SSCp"*))* thiit iiit
I«f» i Woirt((5, ® ftp")),
—i>Rxt(a»® 2 ( p ) ) — » o peP
In 0X8et, tb» horotwrolii^w f j pepw f¥ottlfli suV rowp Into
*r»ttlBl ^ntjfroup and h»nc» th« «oni«fieo}
peP P«P Pfep
i9 oxect.
•3
t« ar* F«QiiiF«d to prevt that •¥ivy 6 In
ii«xt(9, ® E(p)) l8 tilt l»«i?t df soMt 1* in !ir«xt((}, (±) £<n")).
But I5« m att(0 , ® »(t")) wlfita sQcl) tliftt f»i^' • 15. n » o , fiP
1» f 1 • Ktr «• i H«xt(0. (5) 2(p")),
It follcfvfl fipoa tb«er«i<57«1} of { 1 I tliat no
•l«3«nt net In ]loxt(a» ® S(p*)) oaa %• voppod Into tbo piP
Prottlai mitfftwap of Inir ^ » oad ttoRoo
P«P
2 . 9tttgro«ptt And Cnotlont Sroupo*
VIth tho htlp of tbo oxaiet n9cm&nem» dl«ett f<od
In thoerwM<3*2) onA (9*4} « • aro new in a position to
Qonotruol t*o ««Vi; e«p« of NoxtCOt 0 2<p"))f ono oontainod
in tho otti«r and di«oa«»«> tho dooovponition of tbo ciiM>ti«nt
iToup*
fh02£j2j(5.9)*If P itttho diTiaitlo part of 0 , than tho
oxaet aoooanoot
0 — • > D '» •^G > Q ^
u
piP v^p peip
«li«r« a > 1 9 • flB«d intMtr with liesoBarphiffM t
peP p«p pi&p
« I »»xt(0 , ® «(p»))«—.>| |»rt(0, (A) 2(p)) PDF ptiP
PSP *«P
WP P«P
l t«xt (0 , (^ f(p")) / l«rf«S?!I«»(l>, « 2 ( p ) / ® K P ) ) , PitP P6P P6P
Ktrfs/XiAik ff9nt((}/X), 0 Z( p) )(?)»£%<», ® 2(p"*^)). pgP pgfi
Preef>Th» pxaet fi*ni»ne#8 of tli*er«i(91»5) of j 4 {«
thper«*(1.73) •nfl tli«er«M(3«2). (9«4) y l t U %h» follo«lii«
<5
oormitatiTt Almtntsmt
Hrt«l/»,C) 2<p""^ ))-JL.>ftct<0,® f p**" )) PSP
>^
P«P
I .B-l h -—>l5Jrt( »,($)»( p°^')) —->0
• , ^ 1»«xt(«/l>,® Z(p")) a ->H««it(a,® E(p«))
•i !• 0 —>ll«xt(0/ i )^ 2<p))-i—>H»xt<0^ g(p))- l .>
Re«(B,« Z ( p ) / 0 M p ) )
0
p^p pip
0 •>0
with axttcfl ro««« ani oelwrniv*
^Inot t1i« prodvet of two •piperphlffvt f mid g i s
mtmin «a •pivorphiini i t fellevn tli«t fg iw ms tipiP«>rpliiini
h»ne«t
»«att(&, ® 2<p*))/ I«rf« ar !%»<», « Z(p)/ ® Z<p)), m^ p^ WP
Tli« eofi^mtatiTo diairmi and Immm. (S»5) of I 4 I
M
ylald tti« fellflwiiii mto t ••<««i}e«t
W«t(0 / !D^ f ( p" ))® lxt< »,€) 2<P ) ) - - - - - - ->»«xt (0^2( l^» ))
*' . >ft»(P, « S<p) / ($ K(P)) pep ptP
I t s i a n s ;
e«Ffg • Xc *4> D i l l .
iSviacntly •
^Mr 1 Into n IB • ,
peP
«lne« X « K«rff i t f0ll9w« t ^ t |
0 « gx « fsfijt m tdy iMtM> dy « 0»
l a p l l M ,
«T
7ttrth«rMur«t
f t ot'tain two si^fro«p« ! • ** «»* I«P f« of
Ktzt(a, 0 S(p')) 9ia«li that :&r. iilt In oeataintd i s X«r f«
Voiri
Pi*
•ad
Iwh/ XBlilt-»tt(», (t) SCv""^)). 9iF
th* d«ilr«4 rtfmit fellowc*/
2iiS£)^(5*()*Xf (>/b i" • <ir«6t mn 9f t^M agemipa and
$9
«or«lle f^env» of fsriv* (rdfr« thtat
• Ziihic*
1-1
||yyyEj8(5.7).TIi« (wetiMt gro«p K«r f g / I* lilt &•
«et«r9leii*
greef. Sin at dlracft nvm of two oot«r«ieii group* la
oet«r«ion« tlio proof follw* tttm tliooraa(5i*6) of i 4 I
oni flrfl* fropovitien (2*3)«^
9* Coeutotien Of lloxt(f fit)*
In ooetlMi 1 ond 2 wo liovo Ai onsmod tfeo rolo of
iiortCBt^), la OMO A waro tho dlrtet om of corelle igreupo
of pFlao or prl»o pewar ardor* In this ^aotloB vo foaoo our
•ttantloo en tha iproop of K«xt(B,A) , If f 1« tbo dlroot
mm of orelie groaps of peim* er prist ptmm ordtr*
If B 1« th« dlrtet avR of cQrelle groups of prlv«
«rdtr i»« t
B - ^ B.
tlin ttem proponitloa <2«tO)t
• 0,
If B !• tlia dlr»et mm of e^elio jgronpo of prlvo
povar or4or« tlion oi^lii In wim of prepeiiitieB(2»10) i t i«
Mvffieimt to ooii«id«r f o« • flo^elie ireup of prlno pewv
ordtr* Xa tho folloviaf propo«tie» «o di«ting«itti tbo oft«o
whoro A 1« on lafinito mA fiaito ogrolio 0r««Pt rtapoetivoly.
ygopoaitioB(3>9).If ? 1« • cQTO lo group of 9tdm ^
(vboro p i s • iriKO wam\mt\ u > t an intogar aad A in fn
iafinito qrelio groopt tliiot
70
K»xt(T,A) . ZCP**"^).
Proof. l!ro» Uera (52«1fl^) of i 4 I It folle»« tMtf
How, / In an infln4tt oycHc ^ovp thar»fort|
'Py proponltlon (2,3) It follo«« thati
Roxt<P,A) . O p 1?itt{B,A) P«P
• P M P " )
If / In II ejrelle ^roup of t rlff!• ordar, tbin
%9 »«y con^vldw / Ml a oyclie group of prtm» ptmwp wA&ti
It k m Z ( ^ ) , B m f(p")t wliaro p ^q» aro priso
ot»tera» ®nd B»,II > 1 , intftgara tbaej
!laxt(15,/) • 0
71
lBt«ii;«r«« It fellvfttf that I
R#xt(».A) • p^y P lKt<F»A),
• P «(p' ) .
Rene* « • b8Y« fireT«4 th» followiBK firepoiiitioii.
ft^po«it^oB(3«9), If F 111 • fierdie greup of «rd«r p'
•nd / iff « Qsrelitt group of «rdir pF, nhoro p»Q aro prlpt
rni*t«r« and VtB >1 aro inttgors, thin
lf«it(B,i) . ECP*"^) ,
A
»h«ro r « Rln«(B»»)•//
72
th«» ara 9 ' - p' eerelle txttaffleiifi ©f A Vy F, " ine «
egrelie 1^ gjewtv eentalns no aonHriTi^l nest siat i eitiM.
Xt flo1l0wii ttwt til* naat •xt«n<ilonii of ZCp"*) ty E{p^) ar*
•xaetly tlies* •xtmffloiw «hieti art iion«c^oXie«
RaMrk(S>11). fha rtfmlte of tliia efeaptar eaa l9
oarpitd eat aoalegoasly for tha pgftmp af ^ara high
aitanaioaa*
9 ff f y T ^ MT
Btx%_ AiA vf « Orenps*
If « • Wish to find out ft ffovp C tb&t »«9r t«
rt^ftr(l«4i fts thft group of hosovorpliiiicp of • nultftllo
irovp A Vy ft fntltftllo ffrovp V, wo find tho vn^mmt in
(ojroKplo 1 pfturo 1S1 of I 4 i) ftad ot«i«rTO thmt C « Ro*<2«C),
Tf «o think of tlta «if«o proH«n toe th« ifPovp of oxtonttionn,
thftt i^ • vtaftt aro isroopo t)iftt aro fxroupo of oictimwioiio ?
•n OT ifVir iff fiuppliod 1^ ( § 54 propw^ty H of ) 4 i) vbieli
«t«tft« thftt •'wmj rodueod ootorviOB group 9 oon to eonoi«>
dorod fto tbo iproop of oattofiAioiKi of ft auititMo irovp A \j
ft onitftHo i^eup B* lo foot «o hftv« tho a«tiiriil iooworphisB
9 -m Siit((^s:»a),
Za this < tftptar • • oholl ooneontrftto on pttro«4iigh
oxtOBolonn and ««hftll find o«t «;rottpo tli«t oro groopo of
pwo«))iierb oxtonsion** ^^wih greupo will to ofillad
74
H«xt • jcreupt* It wi l l t t ot:?»«rv«d thfct « ^oup irtileh P
ia «lr»et nvm of oydlio «re«P« of tb* ««»• «rd«r p 1<« a
Raxt • group*
Qrevpa all of vlioaa axtanalona %y teralfm ^roupa
apllt ara diirlaltla grovpa* Cafornlon greapa w dafinad
aa froiipa a l l of aboaa axtatwloaa If toralon-firaa groti a
ara ftpllttinf. flMt l9 a group 0 In a oetar«ioB iproup i f
«Kt(CtO) » 0* fu%t 1Sxt(C,9) • Paxt<OvO}« whan C la a toralon*
trtB iiiroap* Zt saana that a f^teMp 0 ia a ooteralea group i f
PaxtCCy?}) m 0. Tlia qaa^tioB whit^ naturally «ii^a«t itnalf
la I vbat m i l ta tha groupa a l l of wteoaa pura axtanaiona
Yj a ^ taraioo i^oap «pllt t Thift auaatieii waa annvw-ad
Vy Harriaoa in i 8 t, and ha oallad aueh graapa high
injaetiTa* Aaaardiag to hia!, a group 0 auoh that o'ai 0
ia high ittjaatiYa (that la 0 ia a auvpand of avarr group
in vhiah i t ia high) i f Paxt(c/Z,a) • 0
Roa fiaouaing an tha Frattini au1 0ra«p of Paxt tha
auaatiofi ariaaa i what ara tha groupa al l of whoaa jmf^
high axtmiaioiiw If twalou groupa «ra aplittiag ? Ta aball
T5
6i««bi« tIdM <i«Mti«i • ! • • IB this dtmp^m m§ slwll « I1
In fPsetiw tf MM VMit iMBM (»• liitp«itt«iit irtiiai
v U l W »••<•# la til* ««will«
9 MB %• rifrt««it«A la th« fbra § • XMelL(BtJi) t <fiMP
«iiit«%l« WHmp9 A mA 3)0 H«it- * «r««9 will %• 4»fl«t4 Mii
i t Hill ym prvrtA tlMt M •l«MBt«»3r p • gremp i« A
•11 •f «liM« 9«r«^ii«li «ct«isi«M ^ t«r«iM grmpt «*•
•plittlsg* a«cli «r««p« will %• m&mA • • 1^ • gt9mp9 A
^m.^ m* « m . t . i t « i K m « . ftrl^ . <r««. M i l 1>.
disoMsaA*
^«fU«B if Vill %• AVfttta %• «i««M«SBff
tilt fr«Hrtit« Af 1^ »0P««9«* It lAU W CMMI
th»t il«MBtcrr f • iTMie MA trnwimf^Of— (r«i9«
•rt 1^ • ^«ttf«*
flMi pwralt t f tliit •luiptflp Art nonnar fiMn |20l
?•
1« ^€m9 Immwm
portHiliaili •xtMAlonn,
Iiw»»(4.1).yor an al«n«ntary p-group Af
Rpcof.^ittca ^ X(q) ooineidts vitk ttui saxle^*! torsion
mitf^onp of « 2(q) and tho faetcr iiroupi
i s AlTiai^a (aaa ttiaoraa 9*2 and asarelao 9«H» roapaetl*
•aly ©f I t4 I )
It l9 tamy to TWify that tha ^otiQapett
oep ^P ^ p qgp
lo puro-hiffh axaot. Thaorae 6 of | 8 | Ipp l i^ t t ^ ' il&i ,>^
\
>HaxtJ ^ H^).A) >0
77
1« •Kscd*
^Inet 9wmfT tUmmtmef p-*irotn» i9« dlrtet 9W of
9 A i d 9Poapt of thft • • • • «r4ir 9 aaA d0«9 net oontain
•l«iiat« •f iaflnitt liMi;^* i t fallows tteat ii* • 0« itlso.
la aiTi9i%l» And hino* ( ^ tt»«ti6& 9 of | 8 {) vt lMiv#t
• 0 ,
*^ine9 A i t m tliai»nt«rj p-grenf {«•• rwark 9*7 )t
t!i«r«for», 'the «*^«fiett
jQOmf.Sifie*!
m9 mf
Hmt 0 l« a d in«i^ l« frouf WIA A i« « ra^etA groap i t
«i»«r«fer« fellawa tltati
« M ( C » A ) • 0.
?BMh«nB«p«, tlia «x«ot»«;fn of tha •anvaaaai
w w f 4i» t i l *
Sff!>Iia« flia ajcaaliia<va af tha •a«i«8eai
Tl
(«•• tliMi*«i(44.4) of 14 !>• Vh9 flrflt group in 0, »iiie«t
iMt group im %h» iP«<!««noo i« 0 sliieo ( 2(o) In • Q group
•nd A i« • p-^roap and limMt
Finally • «t prevo «Qotls$« Immm for •lav^ntury
gropf«ffine>t
90
•ad A' « 0 md C 1« • dlvi t iMt group t i t foUova firoB
I 9 i tliatt
HntjCfA) • Iiaxt(Q,A)
£ laztiCtii)
- 0,
fflBM A ill an tl«i!«fit«ry ir-irmip* Aliio 13r t1taor««(2*1t)|
^ q«P WP '
Bow, < ^ prepooltlon 3.9) oineo Z<o) i9 o torsioa
« HO«t,(E(«),A) • « ( _ p t f«tt<«(o),A)), «|P ' WP **
1« < ,r» q»t»t(«(o),A)),
oiP ^
• 0.
81
Oomtqiitiitlyf tjr l an* (4*1 ) « t hirvt
2 . RtJKt. • Oroapa* r
¥1i« alji of tM« «»etiofi i s to find out gro«p 0
tlwt l« iMvorphle to ttui ^oup of pvroHila^ oxtoawlens
for ovitttMo 9 oop« A oad B* f« shell ooll ««ch froiipo
Btxt • ffroups. r Poflnitioa(4>4).A group 0 la oollod o Hoxt^- group
if.
0 • Resteer »A}
for ouitoVlo groopo A aod B*
(? rollflwing iff ttfi «oia thvoH of tliio flli«pt«r
Tlioegf(4.S)«8vry oloaontopjr p->«reup i« tho Rtxt ogroup.
IWI'td 2, tlio odditivo group of intoforo in tho
diTisit^lo group Qt tho additiTO groups of rational maitara,
that i s , »2 ...£^>Q.
S2
Jmtf ft wmmtmex^wm m t
m 9 m ( « S » ( • • • • • • » • •¥ Q Z f • • • • • • « ) )
vticrt St 2. How proeMtflnf «« «• do In tli«cr«K2«ia) in
clisptir II «• el'tftin tliat cZ in • ntmt nu^getmp of t
SOVf
Q 0 « (Z/<l«) • C011 Z(«l).
»li«r« Z< 4) •t«ad« for Qjrsllo «i o«P of order 9 aotf 7 S in
•MPti««l din Joint tnm 0 2(4)« TMn in in oooordnaoo witli 4«P
tho rnvnrlt of Bnrrinon 1 8 | nnetien 4, thet # in a ntot
nvH itprenp of 0 i f nad onlj i f A in vnziKnl dinjoint froe
noRO nvVioronv K of 0«
It in ontqr to nnt that If2 in • imrn frulgroap of
c 0 « ?(«)• f irttanmerot 0 Hn) in a raxiBal tor<iion
Mtgrottp df % 2<q) vaA thtrtfor* of C ($)M 2 (Q) . Kwett
i« m p»f nidk^fsttnp of
<••• •x«relii« 5» i»ft«t 11« of | 4 {)• How y {Imam H.i of
) 4 I) wt l»v»|
i s • pur* 8nVp>otip of i
froK th*orfli 4 of | 9 |» i t folleoo ttiot tho oo^weot
0 ——>Z X —->Q 0 « E(t)
in puro-^gli oxool
T!)0ortv C of i 8 i ispl iM tliat for m tlMeantary
.>H«jrt fC®« «(q),C> ^ 9iP
Th« firpt irottp 1« 0 \!j Immm (4*2) and th* last
Cro«9 1« 0 \f !«««« (4«3)« Al90»
!hMi(?.»0) ii 0
th«rafer« «• tuRirat
r M&p
Tills proTM t1»at a group nhX^ lo o Alraet mvm of tsrollo
iprovp* of th* 9000 ordor p ia • Roxt • «ro«p«
85
t 3* Rp «» Ore«p««
In tliiff «»etioa «t filiftll di9««fta tli* Ammm ef
group* all 9t wYiMi9 iMr»-4iiK^ •xttf^lom \y totttiw
gttnpB f opllttiB««
^jJ4al||£at.4,«).A grottp i« oftll«« • HJ - «roup i f
•11 itff p«r«<-i)li9i •xttnaieiM ty « torsion i^vap «*•
splitUng. Tlwt i« • ffroup 0 itoftlltd fl^- group i f for
•11 torwioa i ovpo fi
RosEt ( f , 0 ) • 0
«• 4l9eK»« in tli« felleolfig tb«riv • ••••••ary mtA
•ttffieimt eoBAltien for • groop 0 to %o • 81 • gromp*
nk&ormdA»7)»A nooonMry ond iraffloi«it eoa4Uiea for
• <sroop 9 to %o • KJ - «ro«p In thotf
H««* («(P )fO) • 0,
for all irlso nmiVaro p*
»oof>Oyily ffvffieltiiflsr noodo •orlflootioa*
96
9iae» a torsion grofap i t tho 4iroot Mm of i>*gro«p«»
i t iff onffioiMit to proT« tht tUtoroK f«r 9«»ir««i>o» ff T i«
MT p<"ire«p« tlioa »# lurrt tho •xintOBOO of • pw-hif^ oxoet
ooquoneoi
0 ——>H — > f ——>f/B
vitli R dipoot mm of oyolie groups md
1/H .(gSfCpT) •
fho i^uro-liii^ oxoot ooAMiie* yitlAo tlio oxoot «oe»ORoo
HoxtJf/KtO) ——>!l»xt f f , 0 ) «—«»B«xtJI,a) F W W
for any iiroiit ®» ^ ^ iroirp R is torsioai IIOBOO*
Roxt^<R»9) • ^^P f*xt(H,S)
• 0 ,
for R i s tho 4irset svn of eyolie groups* ulsof
Roxt rt/H,0) • RoxtJ ® 2(»*)fS)
87
m !»itt_(Z(p*),0)
0 .
Coii9»(;««Btly,
Htit^(T,0) •©•y/
It i« 9mif to irtrlQr tlwt divinitl* grmipfi sod
•lg«tr«ieally oevpaet groups f S^ • grenps* Alne i t
fell0«« iDRtdiatalj froK th«ort«(2«19) tliait torslen^frM
T)i« flDllciting irop««ltloii anwartff tliftt for • iproHp to
%• • HT • group i t io Ofloiig)! to tihodc ito rodiieod port.
|tfiBfialiail4.8)a groop i« «J • «f«»P o » e t l y , i f ito
rodoood port i««
Proof. l o t ,
a » & ^ i
to tbo dooootpo9ition of tlio group a iato i to diirioillo port
3 ond tbo rodaoad port I* thoo \j tbooroK(2«11) oo hmr»t
Hoxtp(E(p*),0) • Ilo«t^(E(p-), D ® ! )
88
ftmwi9 Air«e% prod«ot «• hsT* th« followlAgt
I^ • «roup i f i»B4 only i f ftft^ ^^^ i s •
fifftcf. fbft proof folloiro fire* tlio fool %1ifttt
H0»t f 2 ( p * ) , « ( » . ) • « Hoxt r I(p*),0. ) ^ i e i * i « i ^ *
Tht follewiBf thoerott fiTow ttero ifiwiidit into HI • ftresiHi*
Thoora«(i.10). lot
0 — • > n — > o - . — M / R ——.>O
to ft pBrt«4iiili oxftot «ft900Beo thoB thft follovliui hel4«
(1) If %ot)i B ftftd O/b ftro 9^ • «ro«pft» t h n no i s 0.
(2) If 9 ift ft HI * «P0«p» thoa 90 in Ht whtBorar tho
fftetor itroup i>» r«duo*4*
m
(5) 0 1« • !|J - itroup i f mftA ©oly If H 1« • 8* <• grmif),
«lifii«vir Q/^ iff t«rflion*&rt«*
0 " i>H I > 0 I > Q ^ • > 0
jrialds t i l* tx«et (••«3ii«Rc«t
fli« firwt and tli« l»«it irevpfi or* Of vine* 8 m4 9/R «r«
H? • gmpfl* Tliar«for«y
•nd 9 i s HiJ • srovp*
i i^^a tli« Htc««iie«t
indnotii tht naot aacmmo*
Tha firwt group 1« 0» ainea «(p**) ia a diipi»i%la gr«BptO/R ia
90
a r»diiMd group* n«o» tbt lust isroup i s 0* tliiM a i« «
R • irottp* Htnet
Htrt^(E(p-),H) .Of
Mift (2) fe l levs .
Hwi(«(p*),0/H) ——>H«x%^(«(p*),H) —•H«t^<2(p'"),0)
— » H « r t y ( «( p*),0/h ) .
If 0 Iff • R » ffpoiipt thtn
r
Also, G/H in • tor9leB<->frt« ^tnp and tip*) in • tersiea
group, lianeo
91
th9ftegp9t
w
If R li9 • BE • coup 9mA O/n in m tor«ioii»fir«« grenp
then ttet 2ii4 MiA 4^^ frotipA in tb* mtevt inaot fi^cpme* art
0« and 0 iff • f • ^ o n p . / /
Rev « • 4iff«n«i« vadir vbiit emi41tieii« e ratfuead iproup
Hill I* » t^ • grmip*
JQ)fS£fi(4«1l)« A rtduotd greap A im R «> grevp If ani
only i f lit<Q/ZtA) i" * 1^ * 0^««P*
iadiio»» tlia •met «*c«iiie»t
iiMi(M) —--^R^2»A) —-^Bxt(c/)S,A) ——>fict<C(»A)
—->««%( Z , A ) ——>0
Thm tirmt grovp i» 0 , also tha last gveup i« 0 sinea Z ia
92
m tt— group. P«rth«r«Bort Ext(C>A) i» • t«r«leB*fk'M
86B(Z»il) S( A •
I f fOll0»« ttlAt tht «*<}««IOt|
0 —.>A ——>teK q/Si ,A) — » ! « % < 0,A ) ——>0
i « v«r« •xaet iMd j r l t l tn t1i« • ia« t ««q«*««t
!lB«(f(f*),l«t<C>A)) »F« t t<2 ( i r ) , * ) —•P««t<2(?*) .««*<C/Z,A))
—^P«xt(2:(fr)t1tlt(09A)) ——»0
TlM f l r t i t and lawt fqrettp« of 1to« •x»et (••iniM»e« a r t 0« Also^
Draf t la i attti^raiipa 9f two iaaiMrpliie grovpa ar* laagerphio
i t fe l laaa t lwt t
To luiTO «ora iaal f l i t into QJ • groapa »a dlnoBai aa
o t h » eharaetarisatioa of fl^ • lproap« tluit ia i
f | iOfyff(4.1g). A la Hp • croup, i f and only i f
]PMt(Z<p'*)»A) iff iMORorplilo to a ««tfraiip of « Z(p)* 9iP
93
Proof* lot A tt o nr • crenpt thin it fellowo thoti
Ho»t^<8(p )»A)-iO*
CoBffldar tht bovoiior^dMii
m I Poxt<«(p ),*) — - > « (PostCXiDtAVv Po«t(2<p-).A))
4ofin«d tyt
« 1 » ( • 1 • p mt(SCfr*),A) )»
vhoro t c 9oxt(Z(p*)9/}«
Vev i f « » • 0 It fellowo tlsot
( t 15 • p F«ct(£(p**),A), ) • 0,
whloli i»plio« thot
»« ptP P ^•»*C2(p"')t^)«
ITov \y propOftitioB (2«9 ) i t fellowii tliat
B«]tox^t(p*)»A) . 0
TboroferOf « i» • nonevoTT ioB*
H
•ntgroap af Pt]i%(z(p*),A)» » t li«v«t
P«tt(2(p*),A)/p P«xt(2(p-),A) « Z(p).
Oonrartttljy l«t
P«xt(3:(p'*)tA) ii to ft smVfpevp of « S(p) thtn, piiP
Trftttini 9iit|Ero«p« of i«o»«ri^i9 froupo wil l to loororphlo,
i t folloiHi that I
Btxt f2(p"'),A) . 0 ,
onA A t ooMOo • ^ * ffrenp.^
? ff ^ r ? IB-Y
Wtmy 9«rt *«Vgrettp Iff e BMit »iib^o«p« %«l th«
0Oirrtr«« i» not mlwBfm tma* If 0 • [a] ^{^j § vlitrt
• Mii t ar* tff «rdar« 9' meA p» r»«^etlT«l3r t p«( P* th«B
%h» ffiftiffrcniy [pa ^ %] 1« ««ftt >>ttt not p9f la Of • • •
BMda |11U
n«B« •DtbvMtieiazw lik« KartMS, "ttalt CBA 9««!)«
h*T« tri«A to ^Bmwitf th« groupa i s irtiieb feotb tlui eoBo«i 8
oolBeid«* !l«««ntly« 91»miti IWT* elfts^ifiti tli« attlitti
gr«09« in wlilQik «¥«r3r BMt sBt grcvp Is p«rB*
fBmiiag to «xt«o«loB9 i t i s oloor frc* dofinitioiio
of i>iiro oMd Boot oxtODOloBA tluit OTory pvro oxtonoioa io
0 Boot oztoimieBt t>«t ovory Boat oxtmoiea aood aet lo o
B«ro ostOBoloa*
If K io o fttft KPovp of 0» tlioB tho »aot «to««OBoot
96
in • f«p«rt«lii«li mrtmsieii i f B 1« WNiiR»l din^lat ftm
t n i (li '¥t)/K 1« pure in 9A« ^nA « t-««ftt«liii^ •xt«a«*
«i0a i f B ill naziffAl 4iBjeint frm t and (B *%)/t i«
«••% itt 0A«
Siaat (B 4> f ) A i« Pttr* in OA iaplits (B • f )A is*
«t«t ia OA i t fello»» iestdiftt*!^ that •vary I«9«r»«tiiffli
•X%»S9i0fl i n • K««M^<-llig|} •3Et«f)9iOfl t m t A t f l« f t ' t« te i« l l
•xtVQiilea ii««d Bot V* • K"pwMgti tKt«ti«ioii«
iBt«r8tiBff QuastiosM I nnAmt 9\m% aeaditiom * K«B««%<-»
hifljh •xt*n«iea i« ft ir«-tNiFt« ifli «ct«ai9ieo «]id aa«t iKtaa*
«i«B ifli • papft asttnfiieat
Tba ranalts of tlila iii»i>t«p ara jrat to a^^aar ia
T«ltaag Jearnal 9t Batlia«fttiaa }18|«
1« Baat - Kigli Bxtaaaieaa*
la tbia paatiea, «a iavantifata tlia oeadiitiaBa
« B 4 « «liieli a XHiaa-l-MiSi aarlaaaiaa \>aooiiai a X«9i»a«4iigli
^^^,loi ,« wa aatatlitfe ia thin Airacstiaa tba fatleviag I
97
jymSI^5*1)« If B i s • 9qoar««tlr«t aatiir&l ni»t«r«
•sA R and I ar« 8«tgroiif>s of Of snoli ttAt(S ^ K) is ft
41r»0l ffw 0f cgrells groups of tli* sas* ordar n, tbaa
(B 4> K)A i« * i»Mt mtgroQp af oA If •>< Mly i f (H • D A
i« • diraot mHfmamA of dA*
Pre f* Xf(R4>E)A i« < ^aftt aultiprcttf af oAt thin i t
fallow* far all p$f%
wMah ia tara i«i acmiTalant tof
wiin •K}A> «H •i)A)n«(oA)
for al l saaara-fpaa natural amtara ft*
XB partieiilar i f « « o «a mat Iwrai
a((R 4> X}A) • H*
((H • l ) A ) n n ( 0 A ) - ! •
Tlia sat af all aa ftroapa af <>A eoBtaiaint B( 8 A )
99
« 4 di«Joist tetm (R • K)A ia B<^ esply «i4 1« laduetiTt*
B«noo tjr 2orB*« IMMI thirt • l i s ts a vsxitral mmt-m stQT BA<
Ttaa B A i« • (R 4> t)A*toi«)> stttirmip of oAt sst )15)«
1st ,
« - P i PjlKj IV 'b
^s tlis fsstsrlsstlen of n into diffwtnt prisss* Isi
tt(f 4^1) § B ( 0 A ) 1 B A
••4 thsrsfiprSf
Pi P2 ^ 'W *\(9^h V*.
Xt fsUsss ^f Imm (9.8) of i 4 t ^itt
Pj f, . . . . . 1^ f^(«4X)«(V • D / ^ ^ B A *
99
Pi 13 ^ '^^-I ^ *
Pi Pa »t Pfl-i
•r» r t le t lTc ly f>rip», h«no« thare • x l t t Integftre
«i^^; ^ S » • «B
fmeh that
^ - «1 P? P3 'V P|i-1 Pn
• Pi «2 P3 ^r P|i-1 «Vi
Pi Pa Pp Pn-I S
!lo« for ( f • X) e o A t « • littwt lisivaf
(g • K) - «^ Pg ^ • . . . f . . . . . . . p^^, |^ (g • K)
* Pi «2 P5 Pp "Vl-I ^i<« * ^^
loe
• M '
• P^ i»2 • •—• V 'Vi-i V « •^>-
^ine* all tiM 9ini« in t1i« aVeT* amiAtlefi Itleng toi
i t fl»ll«»ii that,
SA i ( H • D A ®V*«
TIM r«v«rflt inelufflon fiolloirff trt*m lmmmi^*9) of I 4 i
Bn4 litne«t
OA - (H • D A ^ B A *
It follows tliAt (H 4.1C)A iti • dir^et •im«a4 of OA*
<«iBe« airary dirtot mnraiaod !*> m a«ftt «iift'ipre«ipt( *« P*f« 91
•t I 5 J).
flit Immm 1« eovplttoly prevtd*/
im tb« «»xt tfeiort* vt «!• • tli« oendltien «a4<r «hi«h
• RfiMt-iilsli •xt«n«i«n V«oe«*9 • Emptor••hli^ •sctcaiiieii*
101
2laaBJB(9«t)« I f a ill a ntmv-'tf iMtiiral mnVtr,
ie • I«piv*-Mgli •xtintioB i f (R > r ) in tb« «lr«et »m
• f iqrttlia i{reii?« of tha « « • «rd«» a*
,gQ|fX« ^iBot tilt ntqtiMiMt
0 . _ > B — - > o ——Ml — > 0
i« a K«««at«ia«h txaet, i t folUva that H i« Mxiaal
4 ia^iBt tpm f ana (R 4, E)A i i aaat is aA« FartliarMra»
(R • X) ia tba diraet mm af «grelia iroafa af tha ^ava
flr4ar a*
Vh9 raoairtaaata af laaaa (5*1) ara aatiAfiad and
aa (R 4- K)A i« * diraet a«MM«i4 ^t d/t. Bat arary diract
aapwaad ia a yara aat«r«f«p (praT»irty *a* pafa 114 af I 4 U
i t fellava tliat (H 4>K)A i" * f « r a aaticraav af sA*
Raaaa t in ffa^maaoat
0 >H ——>o ——>B — > 0
i t I-9ara^«te axaet*^
102
In eft«« of Bltm&ntaty p-ip-ottps •» hay* tli# fellowlaisr
intir««tiiif th»or«*.
0 — > H -——>0 •»B -—» 0
ic a l[«pttr««-l)ij;h »xt«n^ion If (H • E) In an tlmmt&ry
gpoef. An tfltmsntery p-srimp i« the aiP»ct mm of
cgrelie froups of tht «(MR« ordtr p» It follow* that (H • l )
Ifi tho direct fitai of eyclie firotipo of th» ntmt witr p*
(5.1) e<»T»l*t9« th» wroof of tho thtorm,//
2« fstot /n^ PUT* Kxt«n«iOfUi*
In this* ttoetloB »« "hisll di»ctt«« t wider what omidltioRv
1«? « noot »tt%pfr©iip • por» wutjrrowp ? Cr In othwr word*,
uiidM* Whet eondloasff do** a noat «xt«if«lon radaea itnalf to
c Dura f%tan«lon ? Tha folloislnf laPi^ <^lw^ an cn*f»ar to
thi-n ma^ivtlon.
JJSS£|[$«4)* If n 1» 8 aicwara-'fr'*© natwrej sue>t«r and
H, a fviib^oiip of « <?rciip 0, «n»oh that H Is « dlraet ^HB of
105
CQrelle gronpfl of %li« •••>• «a A«r n* tlitB R in « ntftt mi^-
urcrsp 6f 0» i f aaA oaljr If H l* « dlr*et •maRiBid of a«
Bfo^f. TlM proof i» KUQix tho 9OT)« o«« tliat of Immm
(5 .1) . /
t—KotS.4). ^ksoirtff tht oonaitlos* undor uhlcfti a
Boot oxtoBffion l9 o piiro ortonalwi ond i s oontoinod la
thoar—(S.S)> If n io o «qiioro"^oo aottiral !»«•%«?»
thon tho iMOt oxtonoloni
•plit9t i f R is o diroet ouai of oyolio grooyo of tlio
ivopo ordop n.
fr—t* Tho ox8otao«« of th« ffft^aneo i»plio« ttaet
n i« o Boot ovVfiprottp of 0. Al«o H iff o diroot ovo of
egrelie f ottpa of tho ««»• ordw B* Tho rooniroBontfi of
iMNio («>.i) oro fi«ti«fiod oBd thorofojTo B io « diroet
of G.
Ronoo tho !«oa«ooooi
•plitff*
to.
9iiio0 tvtry direct svetmrnii in ptir«» tht ff^ruaac* in
pwrt •xaet an K«11*^
Ccnie«rnlii' th* •ltB«at«ry p-*f{rottpi9 ttieor«»(5.5) t«k««
tho following? tlafi-wit for» , /
yh#«Mrtf>(5.6). The naat AKtimtiiofit
0 >H —->0 — > P — > 0
-»TJllt«», If H i s an »l«»»Btery p-^roup./'
RwrmrlcfS.?)* I t fol1ew<t l!nr»dlf»ti»1y that for »ny irroop
I and for an eltmtnteaty p-i!sreitp (eyello ii oup of i^l»«
ortiar) H wa h»vat
f?axt(P,H) • 0 .
for pora-hlarh -xt«ei»lcn« »a hnva tha follcn»iiM;«
PropO'*itiop( 5»S).yop an alamantary p- Krotip # and a
torsion group Pf
Haxt^TtA) «• 0*
Proof. I OB pret>Offitloii( :?.9) It follown that
Raxt^(I<,A) i PaxtCT-tA)
i Waxt(B.A)
Tf T l* an alftiaBtary p-stroop w» hava th^ ^oTlowlmr,
109
ftrep—itloa(9.9). For an •Itnmitery p-^ovp B t
ll«rt((7,A) • 0.
ftrcof. This i s la aooordMiot wltli tli« rMtat of
Bmptn*l«eii I 9 i «He proTtA that i If
\hm f« T(B»A) i f and only If t
p-1 C f(a»ic ) t m A*
CoBaa^antlyt i f B in a cyelie group af prisa «rd«r p,
F*(B,A) • T(B,A) ae that Ilaxt(B»A) • 0.
Far para<4ilgh aztamvleaa wa hava
|E£BSSliiSBl9«10). Far aa alwaatary F-craup
B. Bazt fB,A) • 0./$r
g i ? t 3 ; 9 f t p # ? i y
t 1 I B«MMMllffb,k«
i 3 t «i4M« I .
14 I I iB f la i t t ftt)«liaB • • 1 . 1 .
19 t
l i i
IT I
t!D*Ki
(1f iO)t1 IT* l l f ,
11,A«I«
•in
18 I aB4 fiklk«rtB«A f J M « *
lMa«l«iiiil ««tli«is« MB* • f mtlM. ,191( 1999 )«9<i«9|1.
t OB «h« •%««•%«• t f lrtt%«pi« la ataUiQi frovpa* Clii«ge IlliBtti«,( 190 }t199-«0|«
Q J. t Hlffli «KtiB«laiiii af al>«iliaii gro«p«.A«l» IUiEtll«Afl(i4»3«i •w«iw*Ut( 19i9)319-990.
(u)
fytfridr I«tinp««tiTt(19<7)»
|$e t • t A s«tt m <or«li« «st«ii«Uat VIMP Ara lu l t i * (9 ) I f t (19f r )»
111 I Mutef I * f R M U M I i » t i l t t l iMTj mt wiktHiim ipr«i9«1»fi«i* Mai%lullBiT«9t«fmait 5(195«)»IT-11I.
I l l I fip«ii»^««« I Hi#i 9«m;r«i9« •fii%«llMi %m9Um 0t9m99* l«i««^.«f »ft«iu11«(19i1}* 15T5«13t|.
| f9 I II • I OB I « l d ^ i«%gv««p» t f tlbiAim lai^ar•B.A. «?•«»•• ift^J.Qf «Atli ,1l ,( I f «1),
19i9*m4»
^«it9t«B«ll« ••• latl i . frtaet t89f (19i1 )f49Vi4iO«
i n I ' '»
9Mrirt<^« M i Wtk«tB«A, I t p U n i a c ppofirt iM t f k i | ^ fliib«>
(19€f} ,M0»
I I f I K«plaaMk3r,Z« I Zaf iai t t iifealiaa ir««9«»lltii.af
Siihii«a Brass9 laa.Aftw, lfi«liigflii«1994«
Inl »Ji«0<
|1« i VMlik»«4»A»
iMglisl tAUiiMcfatolstft M b U *
119 I
|M t «foiiVB«ty«imvBik l i«tl i»J*i«l* Z^
121 I
( l 9 8 9 M t o BvipOBr}.
i2» I • • • I mttmmim Hkmrf •f «1»BU«n
tt5 1
In ) Wtmtm^^*
t CliBrft«t«fiB«tlQ8i Bf lat«rsBett«BB • f mBBt BB^iVBB^ of atBllBft iVB(«pB« J.XBA&aa 1 IA«I I *SBB,29,
(19«5)»5f-a«.
I tlw tlio«ry of grooiMiriiB Intro-t«Hl«»»AUjni MA BBO0B»X»O»
Boston ffKMio( 19 ($) •
|t5 I S t M i t i , ! . I OB O%OU«B e>0B»B iM trtdA ortrf moot Bi^grBBf to ft poro flBl>iproii»« OBKBMt •XatH.tftiiT.St .Vwil . n t ( f9i9 ^ 109»110,
Ur)
\H ) SlM«tl«E* I OB lf«lii8)i m%tBt9mpm • £ «l>m0B ygg»<^»» CtMMMt* lf«%h«llliY*9t»
a%iauUM IM«i«« ZUia«A« ^•Matli* 10, Cl9€S},ia<«209.
jit I lilk«Ptl*A* I f9tmUm MiteMPyklo !«•«•• • f adzM ft%«II«i gro«p«« 7Rtt«J*«f ]tetlM»Ta*11,I«.1t(19i1 }t579-977.
STMip df p p u f •xt«i«t«B«* ABB.